Averaging approximation to singularly perturbed nonlinear stochastic wave equations
Lv, Yan; Roberts, A. J.
2012-06-01
An averaging method is applied to derive effective approximation to a singularly perturbed nonlinear stochastic damped wave equation. Small parameter ν > 0 characterizes the singular perturbation, and να, 0 ⩽ α ⩽ 1/2, parametrizes the strength of the noise. Some scaling transformations and the martingale representation theorem yield the effective approximation, a stochastic nonlinear heat equation, for small ν in the sense of distribution.
Stochastic Optimal Prediction with Application to Averaged Euler Equations
Energy Technology Data Exchange (ETDEWEB)
Bell, John [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Chorin, Alexandre J. [Univ. of California, Berkeley, CA (United States); Crutchfield, William [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
2017-04-24
Optimal prediction (OP) methods compensate for a lack of resolution in the numerical solution of complex problems through the use of an invariant measure as a prior measure in the Bayesian sense. In first-order OP, unresolved information is approximated by its conditional expectation with respect to the invariant measure. In higher-order OP, unresolved information is approximated by a stochastic estimator, leading to a system of random or stochastic differential equations. We explain the ideas through a simple example, and then apply them to the solution of Averaged Euler equations in two space dimensions.
Average and deviation for slow-fast stochastic partial differential equations
Wang, W.; Roberts, A. J.
Averaging is an important method to extract effective macroscopic dynamics from complex systems with slow modes and fast modes. This article derives an averaged equation for a class of stochastic partial differential equations without any Lipschitz assumption on the slow modes. The rate of convergence in probability is obtained as a byproduct. Importantly, the stochastic deviation between the original equation and the averaged equation is also studied. A martingale approach proves that the deviation is described by a Gaussian process. This gives an approximation to errors of order O(ɛ) instead of order O(√{ɛ}) attained in previous averaging.
Stochastic partial differential equations
Chow, Pao-Liu
2014-01-01
Preliminaries Introduction Some Examples Brownian Motions and Martingales Stochastic Integrals Stochastic Differential Equations of Itô Type Lévy Processes and Stochastic IntegralsStochastic Differential Equations of Lévy Type Comments Scalar Equations of First Order Introduction Generalized Itô's Formula Linear Stochastic Equations Quasilinear Equations General Remarks Stochastic Parabolic Equations Introduction Preliminaries Solution of Stochastic Heat EquationLinear Equations with Additive Noise Some Regularity Properties Stochastic Reaction-Diffusion Equations Parabolic Equations with Grad
Umut Caglar, Mehmet; Pal, Ranadip
2010-10-01
The central dogma of molecular biology states that ``information cannot be transferred back from protein to either protein or nucleic acid.'' However, this assumption is not exactly correct in most of the cases. There are a lot of feedback loops and interactions between different levels of systems. These types of interactions are hard to analyze due to the lack of data in the cellular level and probabilistic nature of interactions. Probabilistic models like Stochastic Master Equation (SME) or deterministic models like differential equations (DE) can be used to analyze these types of interactions. SME models based on chemical master equation (CME) can provide detailed representation of genetic regulatory system, but their use is restricted by the large data requirements and computational costs of calculations. The differential equations models on the other hand, have low calculation costs and much more adequate to generate control procedures on the system; but they are not adequate to investigate the probabilistic nature of interactions. In this work the success of the mapping between SME and DE is analyzed, and the success of a control policy generated by DE model with respect to SME model is examined. Index Terms--- Stochastic Master Equation models, Differential Equation Models, Control Policy Design, Systems biology
Stochastic Averaging and Stochastic Extremum Seeking
Liu, Shu-Jun
2012-01-01
Stochastic Averaging and Stochastic Extremum Seeking develops methods of mathematical analysis inspired by the interest in reverse engineering and analysis of bacterial convergence by chemotaxis and to apply similar stochastic optimization techniques in other environments. The first half of the text presents significant advances in stochastic averaging theory, necessitated by the fact that existing theorems are restricted to systems with linear growth, globally exponentially stable average models, vanishing stochastic perturbations, and prevent analysis over infinite time horizon. The second half of the text introduces stochastic extremum seeking algorithms for model-free optimization of systems in real time using stochastic perturbations for estimation of their gradients. Both gradient- and Newton-based algorithms are presented, offering the user the choice between the simplicity of implementation (gradient) and the ability to achieve a known, arbitrary convergence rate (Newton). The design of algorithms...
Stochastic nonlinear beam equations
Czech Academy of Sciences Publication Activity Database
Brzezniak, Z.; Maslowski, Bohdan; Seidler, Jan
2005-01-01
Roč. 132, č. 1 (2005), s. 119-149 ISSN 0178-8051 R&D Projects: GA ČR(CZ) GA201/01/1197 Institutional research plan: CEZ:AV0Z10190503 Keywords : stochastic beam equation * stability Subject RIV: BA - General Mathematics Impact factor: 0.896, year: 2005
Stochastic porous media equations
Barbu, Viorel; Röckner, Michael
2016-01-01
Focusing on stochastic porous media equations, this book places an emphasis on existence theorems, asymptotic behavior and ergodic properties of the associated transition semigroup. Stochastic perturbations of the porous media equation have reviously been considered by physicists, but rigorous mathematical existence results have only recently been found. The porous media equation models a number of different physical phenomena, including the flow of an ideal gas and the diffusion of a compressible fluid through porous media, and also thermal propagation in plasma and plasma radiation. Another important application is to a model of the standard self-organized criticality process, called the "sand-pile model" or the "Bak-Tang-Wiesenfeld model". The book will be of interest to PhD students and researchers in mathematics, physics and biology.
Anticipated backward stochastic differential equations
Peng, Shige; Yang, Zhe
2007-01-01
In this paper we discuss new types of differential equations which we call anticipated backward stochastic differential equations (anticipated BSDEs). In these equations the generator includes not only the values of solutions of the present but also the future. We show that these anticipated BSDEs have unique solutions, a comparison theorem for their solutions, and a duality between them and stochastic differential delay equations.
Stochastic symmetries of Wick type stochastic ordinary differential equations
Ünal, Gazanfer
2015-04-01
We consider Wick type stochastic ordinary differential equations with Gaussian white noise. We define the stochastic symmetry transformations and Lie equations in Kondratiev space (S)-1N. We derive the determining system of Wick type stochastic partial differential equations with Gaussian white noise. Stochastic symmetries for stochastic Bernoulli, Riccati and general stochastic linear equation in (S)-1N are obtained. A stochastic version of canonical variables is also introduced.
Approximations of Stochastic Partial Differential Equations
Di Nunno, Giulia; Zhang, Tusheng
2014-01-01
In this paper we show that solutions of stochastic partial differ- ential equations driven by Brownian motion can be approximated by stochastic partial differential equations forced by pure jump noise/random kicks. Applications to stochastic Burgers equations are discussed.
Stochastic partial differential equations
Lototsky, Sergey V
2017-01-01
Taking readers with a basic knowledge of probability and real analysis to the frontiers of a very active research discipline, this textbook provides all the necessary background from functional analysis and the theory of PDEs. It covers the main types of equations (elliptic, hyperbolic and parabolic) and discusses different types of random forcing. The objective is to give the reader the necessary tools to understand the proofs of existing theorems about SPDEs (from other sources) and perhaps even to formulate and prove a few new ones. Most of the material could be covered in about 40 hours of lectures, as long as not too much time is spent on the general discussion of stochastic analysis in infinite dimensions. As the subject of SPDEs is currently making the transition from the research level to that of a graduate or even undergraduate course, the book attempts to present enough exercise material to fill potential exams and homework assignments. Exercises appear throughout and are usually directly connected ...
Averaging of multivalued differential equations
Directory of Open Access Journals (Sweden)
G. Grammel
2003-04-01
Full Text Available Nonlinear multivalued differential equations with slow and fast subsystems are considered. Under transitivity conditions on the fast subsystem, the slow subsystem can be approximated by an averaged multivalued differential equation. The approximation in the Hausdorff sense is of order O(ÃÂµ1/3 as ÃÂµÃ¢Â†Â’0.
Trajectory averaging for stochastic approximation MCMC algorithms
Liang, Faming
2010-10-01
The subject of stochastic approximation was founded by Robbins and Monro [Ann. Math. Statist. 22 (1951) 400-407]. After five decades of continual development, it has developed into an important area in systems control and optimization, and it has also served as a prototype for the development of adaptive algorithms for on-line estimation and control of stochastic systems. Recently, it has been used in statistics with Markov chain Monte Carlo for solving maximum likelihood estimation problems and for general simulation and optimizations. In this paper, we first show that the trajectory averaging estimator is asymptotically efficient for the stochastic approximation MCMC (SAMCMC) algorithm under mild conditions, and then apply this result to the stochastic approximation Monte Carlo algorithm [Liang, Liu and Carroll J. Amer. Statist. Assoc. 102 (2007) 305-320]. The application of the trajectory averaging estimator to other stochastic approximationMCMC algorithms, for example, a stochastic approximation MLE algorithm for missing data problems, is also considered in the paper. © Institute of Mathematical Statistics, 2010.
The stochastic Swift-Hohenberg equation
Gao, Peng
2017-09-01
In this paper, we will study the stochastic Swift-Hohenberg equation. The weak martingale solution, stationary martingale solution, invariant measures, mild solution, large deviation principle and random attractors for the stochastic Swift-Hohenberg equation will be considered.
Maximal stochastic transport in the Lorenz equations
Energy Technology Data Exchange (ETDEWEB)
Agarwal, Sahil, E-mail: sahil.agarwal@yale.edu [Program in Applied Mathematics, Yale University, New Haven (United States); Wettlaufer, J.S., E-mail: john.wettlaufer@yale.edu [Program in Applied Mathematics, Yale University, New Haven (United States); Departments of Geology & Geophysics, Mathematics and Physics, Yale University, New Haven (United States); Mathematical Institute, University of Oxford, Oxford (United Kingdom); Nordita, Royal Institute of Technology and Stockholm University, Stockholm (Sweden)
2016-01-08
We calculate the stochastic upper bounds for the Lorenz equations using an extension of the background method. In analogy with Rayleigh–Bénard convection the upper bounds are for heat transport versus Rayleigh number. As might be expected, the stochastic upper bounds are larger than the deterministic counterpart of Souza and Doering [1], but their variation with noise amplitude exhibits interesting behavior. Below the transition to chaotic dynamics the upper bounds increase monotonically with noise amplitude. However, in the chaotic regime this monotonicity depends on the number of realizations in the ensemble; at a particular Rayleigh number the bound may increase or decrease with noise amplitude. The origin of this behavior is the coupling between the noise and unstable periodic orbits, the degree of which depends on the degree to which the ensemble represents the ergodic set. This is confirmed by examining the close returns plots of the full solutions to the stochastic equations and the numerical convergence of the noise correlations. The numerical convergence of both the ensemble and time averages of the noise correlations is sufficiently slow that it is the limiting aspect of the realization of these bounds. Finally, we note that the full solutions of the stochastic equations demonstrate that the effect of noise is equivalent to the effect of chaos.
Stochastic integration and differential equations
Protter, Philip E
2003-01-01
It has been 15 years since the first edition of Stochastic Integration and Differential Equations, A New Approach appeared, and in those years many other texts on the same subject have been published, often with connections to applications, especially mathematical finance. Yet in spite of the apparent simplicity of approach, none of these books has used the functional analytic method of presenting semimartingales and stochastic integration. Thus a 2nd edition seems worthwhile and timely, though it is no longer appropriate to call it "a new approach". The new edition has several significant changes, most prominently the addition of exercises for solution. These are intended to supplement the text, but lemmas needed in a proof are never relegated to the exercises. Many of the exercises have been tested by graduate students at Purdue and Cornell Universities. Chapter 3 has been completely redone, with a new, more intuitive and simultaneously elementary proof of the fundamental Doob-Meyer decomposition theorem, t...
Numerical methods for stochastic differential equations.
Wilkie, Joshua
2004-01-01
Stochastic differential equations (SDE's) play an important role in physics but existing numerical methods for solving such equations are of low accuracy and poor stability. A general strategy for developing accurate and efficient schemes for solving stochastic equations is outlined here. High-order numerical methods are developed for the integration of stochastic differential equations with strong solutions. We demonstrate the accuracy of the resulting integration schemes by computing the errors in approximate solutions for SDE's which have known exact solutions.
Stochastic Levy Divergence and Maxwell's Equations
Directory of Open Access Journals (Sweden)
B. O. Volkov
2015-01-01
Full Text Available One of the main reasons for interest in the Levy Laplacian and its analogues such as Levy d'Alembertian is a connection of these operators with gauge fields. The theorem proved by Accardi, Gibillisco and Volovich stated that a connection in a bundle over a Euclidean space or over a Minkowski space is a solution of the Yang-Mills equations if and only if the corresponding parallel transport to the connection is a solution of the Laplace equation for the Levy Laplacian or of the d'Alembert equation for the Levy d'Alembertian respectively (see [5, 6]. There are two approaches to define Levy type operators, both of which date back to the original works of Levy [7]. The first is that the Levy Laplacian (or Levy d'Alembertian is defined as an integral functional generated by a special form of the second derivative. This approach is used in the works [5, 6], as well as in the paper [8] of Leandre and Volovich, where stochastic Levy-Laplacian is discussed. Another approach to the Levy Laplacian is defining it as the Cesaro mean of second order derivatives along the family of vectors, which is an orthonormal basis in the Hilbert space. This definition of the Levy Laplacian is used for the description of solutions of the Yang-Mills equations in the paper [10].The present work shows that the definitions of the Levy Laplacian and the Levy d'Alembertian based on Cesaro averaging of the second order directional derivatives can be transferred to the stochastic case. In the article the values of these operators on a stochastic parallel transport associated with a connection (vector potential are found. In this case, unlike the deterministic case and the stochastic case of Levy Laplacian from [8], these values are not equal to zero if the vector potential corresponding to the stochastic parallel transport is a solution of the Maxwell's equations. As a result, two approaches to definition of the Levy Laplacian in the stochastic case give different operators. This
Statistical Methods for Stochastic Differential Equations
Kessler, Mathieu; Sorensen, Michael
2012-01-01
The seventh volume in the SemStat series, Statistical Methods for Stochastic Differential Equations presents current research trends and recent developments in statistical methods for stochastic differential equations. Written to be accessible to both new students and seasoned researchers, each self-contained chapter starts with introductions to the topic at hand and builds gradually towards discussing recent research. The book covers Wiener-driven equations as well as stochastic differential equations with jumps, including continuous-time ARMA processes and COGARCH processes. It presents a sp
Stochastic differential equation model to Prendiville processes
International Nuclear Information System (INIS)
Granita; Bahar, Arifah
2015-01-01
The Prendiville process is another variation of the logistic model which assumes linearly decreasing population growth rate. It is a continuous time Markov chain (CTMC) taking integer values in the finite interval. The continuous time Markov chain can be approximated by stochastic differential equation (SDE). This paper discusses the stochastic differential equation of Prendiville process. The work started with the forward Kolmogorov equation in continuous time Markov chain of Prendiville process. Then it was formulated in the form of a central-difference approximation. The approximation was then used in Fokker-Planck equation in relation to the stochastic differential equation of the Prendiville process. The explicit solution of the Prendiville process was obtained from the stochastic differential equation. Therefore, the mean and variance function of the Prendiville process could be easily found from the explicit solution
Stochastic differential equation model to Prendiville processes
Energy Technology Data Exchange (ETDEWEB)
Granita, E-mail: granitafc@gmail.com [Dept. of Mathematical Science, Universiti Teknologi Malaysia, 81310, Johor Malaysia (Malaysia); Bahar, Arifah [Dept. of Mathematical Science, Universiti Teknologi Malaysia, 81310, Johor Malaysia (Malaysia); UTM Center for Industrial & Applied Mathematics (UTM-CIAM) (Malaysia)
2015-10-22
The Prendiville process is another variation of the logistic model which assumes linearly decreasing population growth rate. It is a continuous time Markov chain (CTMC) taking integer values in the finite interval. The continuous time Markov chain can be approximated by stochastic differential equation (SDE). This paper discusses the stochastic differential equation of Prendiville process. The work started with the forward Kolmogorov equation in continuous time Markov chain of Prendiville process. Then it was formulated in the form of a central-difference approximation. The approximation was then used in Fokker-Planck equation in relation to the stochastic differential equation of the Prendiville process. The explicit solution of the Prendiville process was obtained from the stochastic differential equation. Therefore, the mean and variance function of the Prendiville process could be easily found from the explicit solution.
Ambit processes and stochastic partial differential equations
DEFF Research Database (Denmark)
Barndorff-Nielsen, Ole; Benth, Fred Espen; Veraart, Almut
Ambit processes are general stochastic processes based on stochastic integrals with respect to Lévy bases. Due to their flexible structure, they have great potential for providing realistic models for various applications such as in turbulence and finance. This papers studies the connection between...... ambit processes and solutions to stochastic partial differential equations. We investigate this relationship from two angles: from the Walsh theory of martingale measures and from the viewpoint of the Lévy noise analysis....
Numerical Solution of Heun Equation Via Linear Stochastic Differential Equation
Directory of Open Access Journals (Sweden)
Hamidreza Rezazadeh
2014-05-01
Full Text Available In this paper, we intend to solve special kind of ordinary differential equations which is called Heun equations, by converting to a corresponding stochastic differential equation(S.D.E.. So, we construct a stochastic linear equation system from this equation which its solution is based on computing fundamental matrix of this system and then, this S.D.E. is solved by numerically methods. Moreover, its asymptotic stability and statistical concepts like expectation and variance of solutions are discussed. Finally, the attained solutions of these S.D.E.s compared with exact solution of corresponding differential equations.
Stochastic PDEs and Infinite Horizon Backward Doubly Stochastic Differential Equations
Directory of Open Access Journals (Sweden)
Bo Zhu
2012-01-01
Full Text Available We give a sufficient condition on the coefficients of a class of infinite horizon BDSDEs, under which the infinite horizon BDSDEs have a unique solution for any given square integrable terminal values. We also show continuous dependence theorem and convergence theorem for this kind of equations. A probabilistic interpretation for solutions to a class of stochastic partial differential equations is given.
Derivation of Stochastic Equations for Computational Uncertainties ...
African Journals Online (AJOL)
This paper presents a simple mathematical algorithm or procedure for computing the uncertainties at the various percent of data input, using the stochastic approach of simulating the input variables to compute the output variables. A simple algorithm was used to derive stochastic equations for some selected petrophysical ...
Stochastic effects on the nonlinear Schroedinger equation
International Nuclear Information System (INIS)
Flessas, G P; Leach, P G L; Yannacopoulos, A N
2004-01-01
The aim of this article is to provide a brief review of recent advances in the field of stochastic effects on the nonlinear Schroedinger equation. The article reviews rigorous and perturbative results. (review article)
Bifurcations in stochastic equations with delayed feedback
Gaudreault, Mathieu
The bifurcation diagram of a model stochastic differential equation with delayed feedback is presented. We are motivated by recent research on stochastic effects in models of transcriptional gene regulation. We start from the normal form for a pitchfork bifurcation, and add multiplicative or parametric noise and linear delayed feedback. The latter is sufficient to originate a Hopf bifurcation in that region of parameters in which there is a sufficiently strong negative feedback. We find a sharp bifurcation in parameter space, and define the threshold as the point in which the stationary distribution function p(x) changes from a delta function at the trivial state x = 0 to p( x) ˜ xalpha at small x (with alpha = 1 exactly at threshold). We find that the bifurcation threshold is shifted by fluctuations relative to the deterministic limit by an amount that scales linearly with the noise intensity. Analytical expressions for pitchfork and Hopf bifurcation thresholds are given for the model considered. Our results assume that the delay time tau is small compared to other characteristic time scales, not a significant limitation close to the bifurcation line. A pitchfork bifurcation line is found, the location of which depends on the conditional average , where x(t) is the dynamical variable. This conditional probability incorporates the combined effect of fluctuation correlations and delayed feedback. We also find a Hopf bifurcation line which is obtained by a multiple scale expansion around the oscillatory solution near threshold. We solve the Fokker-Planck equation associated with the slowly varying amplitudes and use it to determine the threshold location. In both cases, the predicted bifurcation lines are in excellent agreement with a direct numerical integration of the governing equations. Contrary to the known case involving no delayed feedback, we show that the stochastic bifurcation lines are shifted relative to the deterministic limit and hence that the
Asymptotic analysis for functional stochastic differential equations
Bao, Jianhai; Yuan, Chenggui
2016-01-01
This brief treats dynamical systems that involve delays and random disturbances. The study is motivated by a wide variety of systems in real life in which random noise has to be taken into consideration and the effect of delays cannot be ignored. Concentrating on such systems that are described by functional stochastic differential equations, this work focuses on the study of large time behavior, in particular, ergodicity. This brief is written for probabilists, applied mathematicians, engineers, and scientists who need to use delay systems and functional stochastic differential equations in their work. Selected topics from the brief can also be used in a graduate level topics course in probability and stochastic processes.
Stochastic differential equations, backward SDEs, partial differential equations
Pardoux, Etienne
2014-01-01
This research monograph presents results to researchers in stochastic calculus, forward and backward stochastic differential equations, connections between diffusion processes and second order partial differential equations (PDEs), and financial mathematics. It pays special attention to the relations between SDEs/BSDEs and second order PDEs under minimal regularity assumptions, and also extends those results to equations with multivalued coefficients. The authors present in particular the theory of reflected SDEs in the above mentioned framework and include exercises at the end of each chapter. Stochastic calculus and stochastic differential equations (SDEs) were first introduced by K. Itô in the 1940s, in order to construct the path of diffusion processes (which are continuous time Markov processes with continuous trajectories taking their values in a finite dimensional vector space or manifold), which had been studied from a more analytic point of view by Kolmogorov in the 1930s. Since then, this topic has...
Backward stochastic differential equations with Young drift
Diehl, Joscha; Zhang, Jianfeng
2016-01-01
We prove via a direct fixpoint argument the well-posedness of backward stochastic differential equations containing an additional drift driven by a path of finite $p$-variation with $p \\in [1,2)$. An application to the Feynman-Kac representation of semilinear rough partial differential equations is given.
Modeling animal movements using stochastic differential equations
Haiganoush K. Preisler; Alan A. Ager; Bruce K. Johnson; John G. Kie
2004-01-01
We describe the use of bivariate stochastic differential equations (SDE) for modeling movements of 216 radiocollared female Rocky Mountain elk at the Starkey Experimental Forest and Range in northeastern Oregon. Spatially and temporally explicit vector fields were estimated using approximating difference equations and nonparametric regression techniques. Estimated...
Stochastic Schroedinger equation from optimal observable evolution
International Nuclear Information System (INIS)
Lacroix, Denis
2007-01-01
In this article, we consider a set of trial wave-functions denoted by vertical bar Q> and an associated set of operators A α which generate transformations connecting those trial states. Using variational principles, we show that we can always obtain a quantum Monte-Carlo method where the quantum evolution of a system is replaced by jumps between density matrices of the form D= vertical bar Q a > b vertical bar, and where the average evolutions of the moments of A α up to a given order k, i.e., α 1 >, α 1 A α 2 >,..., α 1 ...A α k >, are constrained to follow the exact Ehrenfest evolution at each time step along each stochastic trajectory. Then, a set of more and more elaborated stochastic approximations of a quantum problem is obtained which approach the exact solution when more and more constraints are imposed, i.e., when k increases. The Monte-Carlo process might even become exact if the Hamiltonian H applied on the trial state can be written as a polynomial of A α . The formalism makes a natural connection between quantum jumps in Hilbert space and phase-space dynamics. We show that the derivation of stochastic Schroedinger equations can be greatly simplified by taking advantage of the existence of this hierarchy of approximations and its connection to the Ehrenfest theorem. Several examples are illustrated: the free wave-packet expansion, the Kerr oscillator, a generalized version of the Kerr oscillator, as well as interacting bosons or fermions.
Cox, S.G.
2012-01-01
The thesis deals with various aspects of the study of stochastic partial differential equations driven by Gaussian noise. The approach taken is functional analytic rather than probabilistic: the stochastic partial differential equation is interpreted as an ordinary stochastic differential equation
Energy Technology Data Exchange (ETDEWEB)
Paul, Wolfgang; Koeppe, Jeanette [Institut fuer Physik, Martin Luther Universitaet, 06099 Halle (Germany); Grecksch, Wilfried [Institut fuer Mathematik, Martin Luther Universitaet, 06099 Halle (Germany)
2016-07-01
The standard approach to solve a non-relativistic quantum problem is through analytical or numerical solution of the Schroedinger equation. We show a way to go around it. This way is based on the derivation of the Schroedinger equation from conservative diffusion processes and the establishment of (several) stochastic variational principles leading to the Schroedinger equation under the assumption of a kinematics described by Nelson's diffusion processes. Mathematically, the variational principle can be considered as a stochastic optimal control problem linked to the forward-backward stochastic differential equations of Nelson's stochastic mechanics. The Hamilton-Jacobi-Bellmann equation of this control problem is the Schroedinger equation. We present the mathematical background and how to turn it into a numerical scheme for analyzing a quantum system without using the Schroedinger equation and exemplify the approach for a simple 1d problem.
Stochastic differential equations and diffusion processes
Ikeda, N
1989-01-01
Being a systematic treatment of the modern theory of stochastic integrals and stochastic differential equations, the theory is developed within the martingale framework, which was developed by J.L. Doob and which plays an indispensable role in the modern theory of stochastic analysis.A considerable number of corrections and improvements have been made for the second edition of this classic work. In particular, major and substantial changes are in Chapter III and Chapter V where the sections treating excursions of Brownian Motion and the Malliavin Calculus have been expanded and refined. Sectio
Stochastic partial differential equations an introduction
Liu, Wei
2015-01-01
This book provides an introduction to the theory of stochastic partial differential equations (SPDEs) of evolutionary type. SPDEs are one of the main research directions in probability theory with several wide ranging applications. Many types of dynamics with stochastic influence in nature or man-made complex systems can be modelled by such equations. The theory of SPDEs is based both on the theory of deterministic partial differential equations, as well as on modern stochastic analysis. Whilst this volume mainly follows the ‘variational approach’, it also contains a short account on the ‘semigroup (or mild solution) approach’. In particular, the volume contains a complete presentation of the main existence and uniqueness results in the case of locally monotone coefficients. Various types of generalized coercivity conditions are shown to guarantee non-explosion, but also a systematic approach to treat SPDEs with explosion in finite time is developed. It is, so far, the only book where the latter and t...
Stochastic 2-D Navier-Stokes Equation
International Nuclear Information System (INIS)
Menaldi, J.L.; Sritharan, S.S.
2002-01-01
In this paper we prove the existence and uniqueness of strong solutions for the stochastic Navier-Stokes equation in bounded and unbounded domains. These solutions are stochastic analogs of the classical Lions-Prodi solutions to the deterministic Navier-Stokes equation. Local monotonicity of the nonlinearity is exploited to obtain the solutions in a given probability space and this significantly improves the earlier techniques for obtaining strong solutions, which depended on pathwise solutions to the Navier-Stokes martingale problem where the probability space is also obtained as a part of the solution
An introduction to stochastic differential equations
Evans, Lawrence C
2014-01-01
These notes provide a concise introduction to stochastic differential equations and their application to the study of financial markets and as a basis for modeling diverse physical phenomena. They are accessible to non-specialists and make a valuable addition to the collection of texts on the topic. -Srinivasa Varadhan, New York University This is a handy and very useful text for studying stochastic differential equations. There is enough mathematical detail so that the reader can benefit from this introduction with only a basic background in mathematical analysis and probability. -George Papa
Kinetic equations for longitudinal stochastic cooling
International Nuclear Information System (INIS)
Bisognano, J.J.
1980-07-01
A kinetic equation approach to stochastic cooling is presented. Equations for one and two particle distribution functions are derived from the principle of conservation of the number of ensemble systems. The violation of Liouville's theorem is expressed by certain self-interaction terms. The two-particle distribution describes Schottky noise and feedback effects and is analyzed by techniques of the Lenard-Balescu equation for plasmas. The resulting expression for the one particle distribution is of the form of a Fokker-Planck equation. The suppression of Schottky signals for arbitrary machine impedance is discussed in terms of particle correlations
Modelling conjugation with stochastic differential equations
DEFF Research Database (Denmark)
Philipsen, Kirsten Riber; Christiansen, Lasse Engbo; Hasman, Henrik
2010-01-01
using a likelihood-ratio test and Akaike's information criterion. Experiments indicating conjugation on the agar plates selecting for transconjugants motivates the introduction of an extended model, for which conjugation on the agar plate is described in the measurement equation. This model is compared......Conjugation is an important mechanism involved in the transfer of resistance between bacteria. In this article a stochastic differential equation based model consisting of a continuous time state equation and a discrete time measurement equation is introduced to model growth and conjugation of two...
Generalized Langevin Equation Description of Stochastic ...
Indian Academy of Sciences (India)
Generalized Langevin equation for stochastic oscillations of accretion disks. We consider that the particles in the disk are in contact with an isotropic and homoge- neous external medium. The interaction of the particles with the cosmic environment is described by a friction force and a random force. The vertical oscillations ...
Generalized Langevin Equation Description of Stochastic ...
Indian Academy of Sciences (India)
Home; Journals; Journal of Astrophysics and Astronomy; Volume 35; Issue 3. Generalized Langevin Equation Description of Stochastic Oscillations of General Relativistic Disks. Chun Sing Leung Gabriela Mocanu Tiberiu Harko. Part VI: Combined Multi-Waveband Observations Volume 35 Issue 3 September 2014 pp 449- ...
Degenerate parabolic stochastic partial differential equations
Czech Academy of Sciences Publication Activity Database
Hofmanová, Martina
2013-01-01
Roč. 123, č. 12 (2013), s. 4294-4336 ISSN 0304-4149 R&D Projects: GA ČR GAP201/10/0752 Institutional support: RVO:67985556 Keywords : kinetic solutions * degenerate stochastic parabolic equations Subject RIV: BA - General Mathematics Impact factor: 1.046, year: 2013 http://library.utia.cas.cz/separaty/2013/SI/hofmanova-0397241.pdf
Path integral methods for stochastic differential equations.
Chow, Carson C; Buice, Michael A
2015-01-01
Stochastic differential equations (SDEs) have multiple applications in mathematical neuroscience and are notoriously difficult. Here, we give a self-contained pedagogical review of perturbative field theoretic and path integral methods to calculate moments of the probability density function of SDEs. The methods can be extended to high dimensional systems such as networks of coupled neurons and even deterministic systems with quenched disorder.
Potential in stochastic differential equations: novel construction
International Nuclear Information System (INIS)
Ao, P
2004-01-01
There is a whole range of emergent phenomena in a complex network such as robustness, adaptiveness, multiple-equilibrium, hysteresis, oscillation and feedback. Those non-equilibrium behaviours can often be described by a set of stochastic differential equations. One persistent important question is the existence of a potential function. Here we demonstrate that a dynamical structure built into stochastic differential equation allows us to construct such a global optimization potential function. We present an explicit construction procedure to obtain the potential and relevant quantities. In the procedure no reference to the Fokker-Planck equation is needed. The availability of the potential suggests that powerful statistical mechanics tools can be used in nonequilibrium situations. (letter to the editor)
On stochastic differential equations with random delay
International Nuclear Information System (INIS)
Krapivsky, P L; Luck, J M; Mallick, K
2011-01-01
We consider stochastic dynamical systems defined by differential equations with a uniform random time delay. The latter equations are shown to be equivalent to deterministic higher-order differential equations: for an nth-order equation with random delay, the corresponding deterministic equation has order n + 1. We analyze various examples of dynamical systems of this kind, and find a number of unusual behaviors. For instance, for the harmonic oscillator with random delay, the energy grows as exp((3/2) t 2/3 ) in reduced units. We then investigate the effect of introducing a discrete time step ε. At variance with the continuous situation, the discrete random recursion relations thus obtained have intrinsic fluctuations. The crossover between the fluctuating discrete problem and the deterministic continuous one as ε goes to zero is studied in detail on the example of a first-order linear differential equation
Jia, Wantao; Zhu, Weiqiu
2014-03-01
A stochastic averaging method for predicting the response of quasi-partially integrable and non-resonant Hamiltonian systems to combined Gaussian and Poisson white noise excitations is proposed. For the case with r (1
Stochastic differential equations and a biological system
DEFF Research Database (Denmark)
Wang, Chunyan
1994-01-01
. The simulated results are compared with the experimental data, and it is found that the Euler method is the most simple end efficient method for the stochastic growth model considered. Estimation of the parameters of the growth model is based on the stochastic Kalman filter and a continuous Markov process......The purpose of this Ph.D. study is to explore the property of a growth process. The study includes solving and simulating of the growth process which is described in terms of stochastic differential equations. The identification of the growth and variability parameters of the process based...... been developed. Their properties and the relationship between them are discussed. The evolution of a dynamic system or process is usually of great practical interest. In order to simulate the evolution of the process, alternative methods are used to get numerical solutions. In this study, Euler...
Stochastic partial differential equations in turbulence related problems
Chow, P.-L.
1978-01-01
The theory of stochastic partial differential equations (PDEs) and problems relating to turbulence are discussed by employing the theories of Brownian motion and diffusion in infinite dimensions, functional differential equations, and functional integration. Relevant results in probablistic analysis, especially Gaussian measures in function spaces and the theory of stochastic PDEs of Ito type, are taken into account. Linear stochastic PDEs are analyzed through linearized Navier-Stokes equations with a random forcing. Stochastic equations for waves in random media as well as model equations in turbulent transport theory are considered. Markovian models in fully developed turbulence are discussed from a stochastic equation viewpoint.
A stochastic solution of the advective transport equation with uncertainty
International Nuclear Information System (INIS)
Williams, M.M.R.
1991-01-01
A model has been developed for calculating the transport of water-borne radionuclides through layers of porous materials, such as rock or clay. The model is based upon a purely advective transport equation, in which the fluid velocity is a random variable, thereby simulating dispersion in a more realistic manner than the ad hoc introduction of a dispersivity. In addition to a random velocity field, which is an observable physical phenomenon, allowance is made for uncertainty in our knowledge of the parameters which enter the equation, e.g. the retardation coefficient. This too, is assumed to be a random variable and contributes to the stochasticity of the resulting partial differential equation of transport. The stochastic differential equation can be solved analytically and then ensemble averages taken over the associated probability distribution of velocity and retardation coefficient. A method based upon a novel form of the central limit theorem of statistics is employed to obtain tractable solutions of a system consisting of many serial legs of varying properties. One interesting conclusion is that the total flux out of a medium is significantly underestimated by using the deterministic solution with an average transit time compared with that from the stochastically averaged solution. The theory is illustrated numerically for a number of physically relevant cases. (author) 8 figs., 4 tabs., 7 refs
Parameter estimation in stochastic differential equations
Bishwal, Jaya P N
2008-01-01
Parameter estimation in stochastic differential equations and stochastic partial differential equations is the science, art and technology of modelling complex phenomena and making beautiful decisions. The subject has attracted researchers from several areas of mathematics and other related fields like economics and finance. This volume presents the estimation of the unknown parameters in the corresponding continuous models based on continuous and discrete observations and examines extensively maximum likelihood, minimum contrast and Bayesian methods. Useful because of the current availability of high frequency data is the study of refined asymptotic properties of several estimators when the observation time length is large and the observation time interval is small. Also space time white noise driven models, useful for spatial data, and more sophisticated non-Markovian and non-semimartingale models like fractional diffusions that model the long memory phenomena are examined in this volume.
On weak solutions of stochastic differential equations
Czech Academy of Sciences Publication Activity Database
Hofmanová, Martina; Seidler, Jan
2012-01-01
Roč. 30, č. 1 (2012), s. 100-121 ISSN 0736-2994 R&D Projects: GA ČR GAP201/10/0752 Institutional research plan: CEZ:AV0Z10750506 Keywords : stochastic differential equations * weak solutions Subject RIV: BA - General Mathematics Impact factor: 0.303, year: 2012 http://library.utia.cas.cz/separaty/2012/SI/hofmanova-0373626.pdf
Stationary conditions for stochastic differential equations
Adomian, G.; Walker, W. W.
1972-01-01
This is a preliminary study of possible necessary and sufficient conditions to insure stationarity in the solution process for a stochastic differential equation. It indirectly sheds some light on ergodicity properties and shows that the spectral density is generally inadequate as a statistical measure of the solution. Further work is proceeding on a more general theory which gives necessary and sufficient conditions in a form useful for applications.
Linear stochastic degenerate Sobolev equations and applications†
Liaskos, Konstantinos B.; Pantelous, Athanasios A.; Stratis, Ioannis G.
2015-12-01
In this paper, a general class of linear stochastic degenerate Sobolev equations with additive noise is considered. This class of systems is the infinite-dimensional analogue of linear descriptor systems in finite dimensions. Under appropriate assumptions, the mild and strong well-posedness for the initial value problem are studied using elements of the semigroup theory and properties of the stochastic convolution. The final value problem is also examined and it is proved that this is uniquely strongly solvable and the solution is continuously dependent on the final data. Based on the results of the forward and backward problem, the conditions for the exact controllability are investigated for a special but important class of these equations. The abstract results are illustrated by applications in complex media electromagnetics, in the one-dimensional stochastic Dirac equation in the non-relativistic limit and in a potential application in input-output analysis in economics. Dedicated to Professor Grigoris Kalogeropoulos on the occasion of his seventieth birthday.
Modelling Evolutionary Algorithms with Stochastic Differential Equations.
Heredia, Jorge Pérez
2017-11-20
There has been renewed interest in modelling the behaviour of evolutionary algorithms (EAs) by more traditional mathematical objects, such as ordinary differential equations or Markov chains. The advantage is that the analysis becomes greatly facilitated due to the existence of well established methods. However, this typically comes at the cost of disregarding information about the process. Here, we introduce the use of stochastic differential equations (SDEs) for the study of EAs. SDEs can produce simple analytical results for the dynamics of stochastic processes, unlike Markov chains which can produce rigorous but unwieldy expressions about the dynamics. On the other hand, unlike ordinary differential equations (ODEs), they do not discard information about the stochasticity of the process. We show that these are especially suitable for the analysis of fixed budget scenarios and present analogues of the additive and multiplicative drift theorems from runtime analysis. In addition, we derive a new more general multiplicative drift theorem that also covers non-elitist EAs. This theorem simultaneously allows for positive and negative results, providing information on the algorithm's progress even when the problem cannot be optimised efficiently. Finally, we provide results for some well-known heuristics namely Random Walk (RW), Random Local Search (RLS), the (1+1) EA, the Metropolis Algorithm (MA), and the Strong Selection Weak Mutation (SSWM) algorithm.
Causal interpretation of stochastic differential equations
DEFF Research Database (Denmark)
Sokol, Alexander; Hansen, Niels Richard
2014-01-01
We give a causal interpretation of stochastic differential equations (SDEs) by defining the postintervention SDE resulting from an intervention in an SDE. We show that under Lipschitz conditions, the solution to the postintervention SDE is equal to a uniform limit in probability of postintervention...... structural equation models based on the Euler scheme of the original SDE, thus relating our definition to mainstream causal concepts. We prove that when the driving noise in the SDE is a Lévy process, the postintervention distribution is identifiable from the generator of the SDE....
Modelling conjugation with stochastic differential equations.
Philipsen, K R; Christiansen, L E; Hasman, H; Madsen, H
2010-03-07
Conjugation is an important mechanism involved in the transfer of resistance between bacteria. In this article a stochastic differential equation based model consisting of a continuous time state equation and a discrete time measurement equation is introduced to model growth and conjugation of two Enterococcus faecium strains in a rich exhaustible media. The model contains a new expression for a substrate dependent conjugation rate. A maximum likelihood based method is used to estimate the model parameters. Different models including different noise structure for the system and observations are compared using a likelihood-ratio test and Akaike's information criterion. Experiments indicating conjugation on the agar plates selecting for transconjugants motivates the introduction of an extended model, for which conjugation on the agar plate is described in the measurement equation. This model is compared to the model without plate conjugation. The modelling approach described in this article can be applied generally when modelling dynamical systems. 2009 Elsevier Ltd. All rights reserved.
Optimal Control with Partial Information for Stochastic Volterra Equations
Directory of Open Access Journals (Sweden)
Bernt øksendal
2010-01-01
Full Text Available In the first part of the paper we obtain existence and characterizations of an optimal control for a linear quadratic control problem of linear stochastic Volterra equations. In the second part, using the Malliavin calculus approach, we deduce a general maximum principle for optimal control of general stochastic Volterra equations. The result is applied to solve some stochastic control problem for some stochastic delay equations.
Stochastic Averaging of Strongly Nonlinear Oscillators under Poisson White Noise Excitation
Zeng, Y.; Zhu, W. Q.
A stochastic averaging method for single-degree-of-freedom (SDOF) strongly nonlinear oscillators under Poisson white noise excitation is proposed by using the so-called generalized harmonic functions. The stationary averaged generalized Fokker-Planck-Kolmogorov (GFPK) equation is solved by using the classical perturbation method. Then the procedure is applied to estimate the stationary probability density of response of a Duffing-van der Pol oscillator under Poisson white noise excitation. Theoretical results agree well with Monte Carlo simulations.
Multivalued stochastic delay differential equations and related ...
African Journals Online (AJOL)
We study the existence and uniqueness of a solution for the multivalued stochastic differential equation with delay (the multivalued term is of subdifferential type):. dX(t) + aφ (X(t))dt ∍ b(t,X(t), Y(t), Z(t)) dt. ⎨ +σ (t, X (t), Y (t), Z (t)) dW (t), t ∈ (s, T). X(t) = ξ (t - s), t ∈ [s - δ, s]. Specify that in this case the coefficients at time t ...
Evaluating methods for approximating stochastic differential equations.
Brown, Scott D; Ratcliff, Roger; Smith, Philip L
2006-08-01
Models of decision making and response time (RT) are often formulated using stochastic differential equations (SDEs). Researchers often investigate these models using a simple Monte Carlo method based on Euler's method for solving ordinary differential equations. The accuracy of Euler's method is investigated and compared to the performance of more complex simulation methods. The more complex methods for solving SDEs yielded no improvement in accuracy over the Euler method. However, the matrix method proposed by Diederich and Busemeyer (2003) yielded significant improvements. The accuracy of all methods depended critically on the size of the approximating time step. The large (∼10 ms) step sizes often used by psychological researchers resulted in large and systematic errors in evaluating RT distributions.
International Nuclear Information System (INIS)
Silva, Milena Wollmann da; Vilhena, Marco Tullio M.B.; Bodmann, Bardo Ernst J.; Vasques, Richard
2015-01-01
The neutron point kinetics equation, which models the time-dependent behavior of nuclear reactors, is often used to understand the dynamics of nuclear reactor operations. It consists of a system of coupled differential equations that models the interaction between (i) the neutron population; and (II) the concentration of the delayed neutron precursors, which are radioactive isotopes formed in the fission process that decay through neutron emission. These equations are deterministic in nature, and therefore can provide only average values of the modeled populations. However, the actual dynamical process is stochastic: the neutron density and the delayed neutron precursor concentrations vary randomly with time. To address this stochastic behavior, Hayes and Allen have generalized the standard deterministic point kinetics equation. They derived a system of stochastic differential equations that can accurately model the random behavior of the neutron density and the precursor concentrations in a point reactor. Due to the stiffness of these equations, this system was numerically implemented using a stochastic piecewise constant approximation method (Stochastic PCA). Here, we present a study of the influence of stochastic fluctuations on the results of the neutron point kinetics equation. We reproduce the stochastic formulation introduced by Hayes and Allen and compute Monte Carlo numerical results for examples with constant and time-dependent reactivity, comparing these results with stochastic and deterministic methods found in the literature. Moreover, we introduce a modified version of the stochastic method to obtain a non-stiff solution, analogue to a previously derived deterministic approach. (author)
Energy Technology Data Exchange (ETDEWEB)
Silva, Milena Wollmann da; Vilhena, Marco Tullio M.B.; Bodmann, Bardo Ernst J.; Vasques, Richard, E-mail: milena.wollmann@ufrgs.br, E-mail: vilhena@mat.ufrgs.br, E-mail: bardobodmann@ufrgs.br, E-mail: richard.vasques@fulbrightmail.org [Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, RS (Brazil). Programa de Pos-Graduacao em Engenharia Mecanica
2015-07-01
The neutron point kinetics equation, which models the time-dependent behavior of nuclear reactors, is often used to understand the dynamics of nuclear reactor operations. It consists of a system of coupled differential equations that models the interaction between (i) the neutron population; and (II) the concentration of the delayed neutron precursors, which are radioactive isotopes formed in the fission process that decay through neutron emission. These equations are deterministic in nature, and therefore can provide only average values of the modeled populations. However, the actual dynamical process is stochastic: the neutron density and the delayed neutron precursor concentrations vary randomly with time. To address this stochastic behavior, Hayes and Allen have generalized the standard deterministic point kinetics equation. They derived a system of stochastic differential equations that can accurately model the random behavior of the neutron density and the precursor concentrations in a point reactor. Due to the stiffness of these equations, this system was numerically implemented using a stochastic piecewise constant approximation method (Stochastic PCA). Here, we present a study of the influence of stochastic fluctuations on the results of the neutron point kinetics equation. We reproduce the stochastic formulation introduced by Hayes and Allen and compute Monte Carlo numerical results for examples with constant and time-dependent reactivity, comparing these results with stochastic and deterministic methods found in the literature. Moreover, we introduce a modified version of the stochastic method to obtain a non-stiff solution, analogue to a previously derived deterministic approach. (author)
Stochastic reliability analysis using Fokker Planck equations
International Nuclear Information System (INIS)
Hari Prasad, M.; Rami Reddy, G.; Srividya, A.; Verma, A.K.
2011-01-01
The Fokker-Planck equation describes the time evolution of the probability density function of the velocity of a particle, and can be generalized to other observables as well. It is also known as the Kolmogorov forward equation (diffusion). Hence, for any process, which evolves with time, the probability density function as a function of time can be represented with Fokker-Planck equation. In stochastic reliability analysis one is more interested in finding out the reliability or failure probability of the components or structures as a function of time rather than instantaneous failure probabilities. In this analysis the variables are represented with random processes instead of random variables. A random processes can be either stationary or non stationary. If the random process is stationary then the failure probability doesn't change with time where as in the case of non stationary processes the failure probability changes with time. In the present paper Fokker Planck equations have been used to find out the probability density function of the non stationary random processes. In this paper a flow chart has been provided which describes step by step process for carrying out stochastic reliability analysis using Fokker-Planck equations. As a first step one has to identify the failure function as a function of random processes. Then one has to solve the Fokker-Planck equation for each random process. In this paper the Fokker-Planck equation has been solved by using Finite difference method. As a result one gets the probability density values of the random process in the sample space as well as time space. Later at each time step appropriate probability distribution has to be identified based on the available probability density values. For checking the better fitness of the data Kolmogorov-Smirnov Goodness of fit test has been performed. In this way one can find out the distribution of the random process at each time step. Once one has the probability distribution
Invariant measures for stochastic nonlinear beam and wave equations
Czech Academy of Sciences Publication Activity Database
Brzezniak, Z.; Ondreját, Martin; Seidler, Jan
2016-01-01
Roč. 260, č. 5 (2016), s. 4157-4179 ISSN 0022-0396 R&D Projects: GA ČR GAP201/10/0752 Institutional support: RVO:67985556 Keywords : stochastic partial differential equation * stochastic beam equation * stochastic wave equation * invariant measure Subject RIV: BA - General Mathematics Impact factor: 1.988, year: 2016 http://library.utia.cas.cz/separaty/2016/SI/ondrejat-0453412.pdf
Mixed effects in stochastic differential equation models
DEFF Research Database (Denmark)
Ditlevsen, Susanne; De Gaetano, Andrea
2005-01-01
maximum likelihood; pharmacokinetics; population estimates; random effects; repeated measurements; stochastic processes......maximum likelihood; pharmacokinetics; population estimates; random effects; repeated measurements; stochastic processes...
Lyapunov functionals and stability of stochastic functional differential equations
Shaikhet, Leonid
2013-01-01
Stability conditions for functional differential equations can be obtained using Lyapunov functionals. Lyapunov Functionals and Stability of Stochastic Functional Differential Equations describes the general method of construction of Lyapunov functionals to investigate the stability of differential equations with delays. This work continues and complements the author’s previous book Lyapunov Functionals and Stability of Stochastic Difference Equations, where this method is described for discrete- and continuous-time difference equations. The text begins with a description of the peculiarities of deterministic and stochastic functional differential equations. There follow basic definitions for stability theory of stochastic hereditary systems, and a formal procedure of Lyapunov functionals construction is presented. Stability investigation is conducted for stochastic linear and nonlinear differential equations with constant and distributed delays. The proposed method is used for stability investigation of di...
Stochastic Levy Laplacian and d'Alambertian and Maxwell's equations
Directory of Open Access Journals (Sweden)
B. O. Volkov
2015-01-01
Full Text Available One of the main reasons for interest in the Levy Laplacian and its analogues such as Levy d'Alembertian is a connection of these operators with gauge fields. The theorem proved by Accardi, Gibillisco and Volovich stated that a connection in a bundle over a Euclidean space or over a Minkowski space is a solution of the Yang-Mills equations if and only if the corresponding parallel transport to the connection is a solution of the Laplace equation for the Levy Laplacian or of the d'Alembert equation for the Levy d'Alembertian respectively (see [5, 6]. There are two approaches to define Levy type operators, both of which date back to the original works of Levy (see [7]. The first is that the Levy Laplacian (or Levy d'Alembertian is defined as an integral functional generated by a special form of the second derivative. This approach is used in the works [5, 6], as well as in the paper [8] of Leandre and Volovich, where stochastic Levy-Laplacian is discussed. Another approach to the Levy Laplacian is defining it as the Cesaro mean of second order derivatives along the family of vectors, which is an orthonormal basis in the Hilbert space. This definition of the Levy Laplacian is used for the description of solutions of the Yang-Mills equations in the paper [10].The present work shows that the definitions of the Levy Laplacian and the Levy d'Alembertian based on Cesaro averaging of the second order directional derivatives can be transferred to the stochastic case. In the article the values of these operators on a stochastic parallel transport associated with a connection (vector potential are found. In this case, unlike the deterministic case and the stochastic case of Levy Laplacian from [8], these values are not equal to zero if the vector potential corresponding to the stochastic parallel transport is a solution of the Maxwell's equations. As a result, two approaches to definition of the Levy Laplacian in the stochastic case give different operators
Patchwork sampling of stochastic differential equations.
Kürsten, Rüdiger; Behn, Ulrich
2016-03-01
We propose a method to sample stationary properties of solutions of stochastic differential equations, which is accurate and efficient if there are rarely visited regions or rare transitions between distinct regions of the state space. The method is based on a complete, nonoverlapping partition of the state space into patches on which the stochastic process is ergodic. On each of these patches we run simulations of the process strictly truncated to the corresponding patch, which allows effective simulations also in rarely visited regions. The correct weight for each patch is obtained by counting the attempted transitions between all different patches. The results are patchworked to cover the whole state space. We extend the concept of truncated Markov chains which is originally formulated for processes which obey detailed balance to processes not fulfilling detailed balance. The method is illustrated by three examples, describing the one-dimensional diffusion of an overdamped particle in a double-well potential, a system of many globally coupled overdamped particles in double-well potentials subject to additive Gaussian white noise, and the overdamped motion of a particle on the circle in a periodic potential subject to a deterministic drift and additive noise. In an appendix we explain how other well-known Markov chain Monte Carlo algorithms can be related to truncated Markov chains.
Efficient Estimating Functions for Stochastic Differential Equations
DEFF Research Database (Denmark)
Jakobsen, Nina Munkholt
The overall topic of this thesis is approximate martingale estimating function-based estimationfor solutions of stochastic differential equations, sampled at high frequency. Focuslies on the asymptotic properties of the estimators. The first part of the thesis deals with diffusions observed over...... a fixed time interval. Rate optimal and effcient estimators areobtained for a one-dimensional diffusion parameter. Stable convergence in distribution isused to achieve a practically applicable Gaussian limit distribution for suitably normalisedestimators. In a simulation example, the limit distributions...... multidimensional parameter. Conditions for rate optimality and effciency of estimatorsof drift-jump and diffusion parameters are given in some special cases. Theseconditions are found to extend the pre-existing conditions applicable to continuous diffusions,and impose much stronger requirements on the estimating...
Stochastic differential equations used to model conjugation
DEFF Research Database (Denmark)
Philipsen, Kirsten Riber; Christiansen, Lasse Engbo
be split into measurement noise and system noise. The system noise is used to compensate for those biological processes not explicitly described by the model. Many authors model conjugation by a simple mass action model first proposed by Levin et al. (1979). Also Michaelis-Menten dependence...... by an experiment conducted with E. faecium. In addition, we suggest that a 3rd order time-delay must be included in the model to account for the delay before a newly conjugated plasmid is expressed. A ML estimate of the parameters based on experimental data is found using the software CTSM. The conjugation rate......Stochastic differential equations (SDEs) are used to model horizontal transfer of antibiotic resis- tance by conjugation. The model describes the concentration of donor, recipient, transconjugants and substrate. The strength of the SDE model over the traditional ODE models is that the noise can...
Modeling and Prediction Using Stochastic Differential Equations
DEFF Research Database (Denmark)
Juhl, Rune; Møller, Jan Kloppenborg; Jørgensen, John Bagterp
2016-01-01
deterministic and can predict the future perfectly. A more realistic approach would be to allow for randomness in the model due to e.g., the model be too simple or errors in input. We describe a modeling and prediction setup which better reflects reality and suggests stochastic differential equations (SDEs......) for modeling and forecasting. It is argued that this gives models and predictions which better reflect reality. The SDE approach also offers a more adequate framework for modeling and a number of efficient tools for model building. A software package (CTSM-R) for SDE-based modeling is briefly described....... that describes the variation between subjects. The ODE setup implies that the variation for a single subject is described by a single parameter (or vector), namely the variance (covariance) of the residuals. Furthermore the prediction of the states is given as the solution to the ODEs and hence assumed...
Optimal control of stochastic difference Volterra equations an introduction
Shaikhet, Leonid
2015-01-01
This book showcases a subclass of hereditary systems, that is, systems with behaviour depending not only on their current state but also on their past history; it is an introduction to the mathematical theory of optimal control for stochastic difference Volterra equations of neutral type. As such, it will be of much interest to researchers interested in modelling processes in physics, mechanics, automatic regulation, economics and finance, biology, sociology and medicine for all of which such equations are very popular tools. The text deals with problems of optimal control such as meeting given performance criteria, and stabilization, extending them to neutral stochastic difference Volterra equations. In particular, it contrasts the difference analogues of solutions to optimal control and optimal estimation problems for stochastic integral Volterra equations with optimal solutions for corresponding problems in stochastic difference Volterra equations. Optimal Control of Stochastic Difference Volterra Equation...
Numerical analysis of systems of ordinary and stochastic differential equations
Artemiev, S S
1997-01-01
This text deals with numerical analysis of systems of both ordinary and stochastic differential equations. It covers numerical solution problems of the Cauchy problem for stiff ordinary differential equations (ODE) systems by Rosenbrock-type methods (RTMs).
Nonparametric Bayesian drift estimation for multidimensional stochastic differential equations
Gugushvili, S.; Spreij, P.
2014-01-01
We consider nonparametric Bayesian estimation of the drift coefficient of a multidimensional stochastic differential equation from discrete-time observations on the solution of this equation. Under suitable regularity conditions, we establish posterior consistency in this context.
On Volatility Induced Stationarity for Stochastic Differential Equations
J.M.P., Albin; Bjarne, Astrup Jensen; Anders, Muszta; Martin, Richter
2006-01-01
This article deals with stochastic differential equations with volatility induced stationarity. We study of theoretical properties of such equations, as well as numerical aspects, together with a detailed study of three examples.
Attractors for stochastic strongly damped plate equations with additive noise
Directory of Open Access Journals (Sweden)
Wenjun Ma
2013-04-01
Full Text Available We study the asymptotic behavior of stochastic plate equations with homogeneous Neumann boundary conditions. We show the existence of an attractor for the random dynamical system associated with the equation.
Averaging Einstein's equations : The linearized case
Stoeger, William R.; Helmi, Amina; Torres, Diego F.
We introduce a simple and straightforward averaging procedure, which is a generalization of one which is commonly used in electrodynamics, and show that it possesses all the characteristics we require for linearized averaging in general relativity and cosmology for weak-field and perturbed FLRW
Numerical methods for stochastic partial differential equations with white noise
Zhang, Zhongqiang
2017-01-01
This book covers numerical methods for stochastic partial differential equations with white noise using the framework of Wong-Zakai approximation. The book begins with some motivational and background material in the introductory chapters and is divided into three parts. Part I covers numerical stochastic ordinary differential equations. Here the authors start with numerical methods for SDEs with delay using the Wong-Zakai approximation and finite difference in time. Part II covers temporal white noise. Here the authors consider SPDEs as PDEs driven by white noise, where discretization of white noise (Brownian motion) leads to PDEs with smooth noise, which can then be treated by numerical methods for PDEs. In this part, recursive algorithms based on Wiener chaos expansion and stochastic collocation methods are presented for linear stochastic advection-diffusion-reaction equations. In addition, stochastic Euler equations are exploited as an application of stochastic collocation methods, where a numerical compa...
Development of a texture averaged Reynolds equation
Kraker, A. de; Ostayen, R.A.J. van; Rixen, D.J.
2010-01-01
The application of textured bearing surfaces results in a more complex lubricant flow pattern compared to smooth bearing surfaces. In order to capture the more complex flow pattern and possible inertia effects in the vicinity of the surface pockets, the NavierStokes equations should be used to model
Perturbative approach to non-Markovian stochastic Schroedinger equations
International Nuclear Information System (INIS)
Gambetta, Jay; Wiseman, H.M.
2002-01-01
In this paper we present a perturbative procedure that allows one to numerically solve diffusive non-Markovian stochastic Schroedinger equations, for a wide range of memory functions. To illustrate this procedure numerical results are presented for a classically driven two-level atom immersed in an environment with a simple memory function. It is observed that as the order of the perturbation is increased the numerical results for the ensemble average state ρ red (t) approach the exact reduced state found via Imamog-barlu ' s enlarged system method [Phys. Rev. A 50, 3650 (1994)
The Logistic Equation and a Linear Stochastic Resonance
International Nuclear Information System (INIS)
Gora, P.F.
2004-01-01
We show analytically that a collective action of two correlated Gaussian white noises, a constant external forcing and a periodic signal leads to a linear stochastic resonance (LSR). The resonance persists for long times, survives averaging over the initial phase of the signal and is characterized by a clear maximum of the signal-to-noise ratio (SNR), unlike other cases of the LSR reported previously. We show that the problem of the LSR is closely related to the behavior of a generalized noisy logistic equation. (author)
On Volatility Induced Stationarity for Stochastic Differential Equations
DEFF Research Database (Denmark)
Albin, J.M.P.; Astrup Jensen, Bjarne; Muszta, Anders
2006-01-01
This article deals with stochastic differential equations with volatility induced stationarity. We study of theoretical properties of such equations, as well as numerical aspects, together with a detailed study of three examples.......This article deals with stochastic differential equations with volatility induced stationarity. We study of theoretical properties of such equations, as well as numerical aspects, together with a detailed study of three examples....
Symmetries of stochastic differential equations: A geometric approach
Energy Technology Data Exchange (ETDEWEB)
De Vecchi, Francesco C., E-mail: francesco.devecchi@unimi.it; Ugolini, Stefania, E-mail: stefania.ugolini@unimi.it [Dipartimento di Matematica, Università degli Studi di Milano, via Saldini 50, Milano (Italy); Morando, Paola, E-mail: paola.morando@unimi.it [DISAA, Università degli Studi di Milano, via Celoria 2, Milano (Italy)
2016-06-15
A new notion of stochastic transformation is proposed and applied to the study of both weak and strong symmetries of stochastic differential equations (SDEs). The correspondence between an algebra of weak symmetries for a given SDE and an algebra of strong symmetries for a modified SDE is proved under suitable regularity assumptions. This general approach is applied to a stochastic version of a two dimensional symmetric ordinary differential equation and to the case of two dimensional Brownian motion.
Numerical Analysis for Stochastic Partial Differential Delay Equations with Jumps
Li, Yan; Hu, Junhao
2013-01-01
We investigate the convergence rate of Euler-Maruyama method for a class of stochastic partial differential delay equations driven by both Brownian motion and Poisson point processes. We discretize in space by a Galerkin method and in time by using a stochastic exponential integrator. We generalize some results of Bao et al. (2011) and Jacob et al. (2009) in finite dimensions to a class of stochastic partial differential delay equations with jumps in infinite dimensions.
Stochastic differential equations and numerical simulation for pedestrians
Energy Technology Data Exchange (ETDEWEB)
Garrison, J.C.
1993-07-27
The mathematical foundation of the Ito interpretation of stochastic ordinary and partial differential equations is briefly explained. This provides the basis for a review of simple difference approximations to stochastic differential equations. An example arising in the theory of optical switching is discussed.
On Some Fractional Stochastic Integrodifferential Equations in Hilbert Space
Directory of Open Access Journals (Sweden)
Hamdy M. Ahmed
2009-01-01
Full Text Available We study a class of fractional stochastic integrodifferential equations considered in a real Hilbert space. The existence and uniqueness of the Mild solutions of the considered problem is also studied. We also give an application for stochastic integropartial differential equations of fractional order.
Stochastic equations for complex systems theoretical and computational topics
Bessaih, Hakima
2015-01-01
Mathematical analyses and computational predictions of the behavior of complex systems are needed to effectively deal with weather and climate predictions, for example, and the optimal design of technical processes. Given the random nature of such systems and the recognized relevance of randomness, the equations used to describe such systems usually need to involve stochastics. The basic goal of this book is to introduce the mathematics and application of stochastic equations used for the modeling of complex systems. A first focus is on the introduction to different topics in mathematical analysis. A second focus is on the application of mathematical tools to the analysis of stochastic equations. A third focus is on the development and application of stochastic methods to simulate turbulent flows as seen in reality. This book is primarily oriented towards mathematics and engineering PhD students, young and experienced researchers, and professionals working in the area of stochastic differential equations ...
Neutral Backward Stochastic Functional Differential Equations and Their Application
Wei, Wenning
2013-01-01
In this paper we are concerned with a new type of backward equations with anticipation which we call neutral backward stochastic functional differential equations. We obtain the existence and uniqueness and prove a comparison theorem. As an application, we discuss the optimal control of neutral stochastic functional differential equations, establish a Pontryagin maximum principle, and give an explicit optimal value for the linear optimal control.
Backward stochastic differential equations from linear to fully nonlinear theory
Zhang, Jianfeng
2017-01-01
This book provides a systematic and accessible approach to stochastic differential equations, backward stochastic differential equations, and their connection with partial differential equations, as well as the recent development of the fully nonlinear theory, including nonlinear expectation, second order backward stochastic differential equations, and path dependent partial differential equations. Their main applications and numerical algorithms, as well as many exercises, are included. The book focuses on ideas and clarity, with most results having been solved from scratch and most theories being motivated from applications. It can be considered a starting point for junior researchers in the field, and can serve as a textbook for a two-semester graduate course in probability theory and stochastic analysis. It is also accessible for graduate students majoring in financial engineering.
Controlling roughening processes in the stochastic Kuramoto-Sivashinsky equation
Kalliadasis, Serafim; Gomes, Susana; Papageorgiou, Demetrios; Pavliotis, Greg; Pradas, Marc
2017-11-01
We present a novel methodology to control the roughening processes of semilinear parabolic stochastic partial differential equations in one dimension, which we exemplify with the stochastic Kuramoto-Sivashinsky equation. The original equation is split into a linear stochastic and a nonlinear deterministic equation so that we can apply linear feedback control methods. Our control strategy is then based on two steps: first, stabilize the zero solution of the deterministic part and, second, control the roughness of the stochastic linear equation. We consider both periodic controls and point actuated ones, observing in all cases that the second moment of the solution evolves in time according to a power-law until it saturates at the desired controlled value. Furthermore, our control framework allows us to force the interfaces to have a prescribed shape. We observe from our numerical experiments that our results are valid for different types of nonlinearity (in particular, the Burgers and KPZ ones) as well as white and coloured noise.
Parameter Estimation in Stochastic Differential Equations; An Overview
DEFF Research Database (Denmark)
Nielsen, Jan Nygaard; Madsen, Henrik; Young, P. C.
2000-01-01
This paper presents an overview of the progress of research on parameter estimation methods for stochastic differential equations (mostly in the sense of Ito calculus) over the period 1981-1999. These are considered both without measurement noise and with measurement noise, where the discretely...... observed stochastic differential equations are embedded in a continuous-discrete time state space model. Every attempts has been made to include results from other scientific disciplines. Maximum likelihood estimation of parameters in nonlinear stochastic differential equations is in general not possible...
Almost Automorphic Solutions to Nonautonomous Stochastic Functional Integrodifferential Equations
Directory of Open Access Journals (Sweden)
Li Xi-liang
2013-01-01
Full Text Available This paper concerns the square-mean almost automorphic solutions to a class of abstract semilinear nonautonomous functional integrodifferential stochastic evolution equations in real separable Hilbert spaces. Using the so-called “Acquistapace-Terreni” conditions and Banach contraction principle, the existence, uniqueness, and asymptotical stability results of square-mean almost automorphic mild solutions to such stochastic equations are established. As an application, square-mean almost automorphic solution to a concrete nonautonomous integro-differential stochastic evolution equation is analyzed to illustrate our abstract results.
PC analysis of stochastic differential equations driven by Wiener noise
Le Maitre, Olivier
2015-03-01
A polynomial chaos (PC) analysis with stochastic expansion coefficients is proposed for stochastic differential equations driven by additive or multiplicative Wiener noise. It is shown that for this setting, a Galerkin formalism naturally leads to the definition of a hierarchy of stochastic differential equations governing the evolution of the PC modes. Under the mild assumption that the Wiener and uncertain parameters can be treated as independent random variables, it is also shown that the Galerkin formalism naturally separates parametric uncertainty and stochastic forcing dependences. This enables us to perform an orthogonal decomposition of the process variance, and consequently identify contributions arising from the uncertainty in parameters, the stochastic forcing, and a coupled term. Insight gained from this decomposition is illustrated in light of implementation to simplified linear and non-linear problems; the case of a stochastic bifurcation is also considered.
1/f Noise from nonlinear stochastic differential equations.
Ruseckas, J; Kaulakys, B
2010-03-01
We consider a class of nonlinear stochastic differential equations, giving the power-law behavior of the power spectral density in any desirably wide range of frequency. Such equations were obtained starting from the point process models of 1/fbeta noise. In this article the power-law behavior of spectrum is derived directly from the stochastic differential equations, without using the point process models. The analysis reveals that the power spectrum may be represented as a sum of the Lorentzian spectra. Such a derivation provides additional justification of equations, expands the class of equations generating 1/fbeta noise, and provides further insights into the origin of 1/fbeta noise.
Classical and stochastic L\\"owner-Kufarev equations
Bracci, Filippo; Contreras, Manuel D.; Díaz-Madrigal, Santiago; Vasil'ev, Alexander
2013-01-01
In this paper we present a historical and scientific account of the development of the theory of the L\\"owner-Kufarev classical and stochastic equations spanning the 90-year period from the seminal paper by K.L\\"owner in 1923 to recent generalizations and stochastic versions and their relations to conformal field theory.
Solution properties of a 3D stochastic Euler fluid equation
Crisan, Dan; Flandoli, Franco; Holm, Darryl D.
2017-01-01
We prove local well posedness in regular spaces and a Beale-Kato-Majda blow-up criterion for a recently derived stochastic model of the 3D Euler fluid equation for incompressible flow. This model describes incompressible fluid motions whose Lagrangian particle paths follow a stochastic process with cylindrical noise and also satisfy Newton's 2nd Law in every Lagrangian domain.
Linear stochastic differential equations with anticipating initial conditions
DEFF Research Database (Denmark)
Khalifa, Narjess; Kuo, Hui-Hsiung; Ouerdiane, Habib
In this paper we use the new stochastic integral introduced by Ayed and Kuo (2008) and the results obtained by Kuo et al. (2012b) to find a solution to a drift-free linear stochastic differential equation with anticipating initial condition. Our solution is based on well-known results from...
Almost Periodic Solutions for Impulsive Fractional Stochastic Evolution Equations
Directory of Open Access Journals (Sweden)
Toufik Guendouzi
2014-08-01
Full Text Available In this paper, we consider the existence of square-mean piecewise almost periodic solutions for impulsive fractional stochastic evolution equations involving Caputo fractional derivative. The main results are obtained by means of the theory of operators semi-group, fractional calculus, fixed point technique and stochastic analysis theory and methods adopted directly from deterministic fractional equations. Some known results are improved and generalized.
Stochastic functional differential equations and sensitivity to their initial path
Baños, David R.; Di Nunno, Giulia; Haferkorn, Hannes; Proske, Frank
2017-01-01
We consider systems with memory represented by stochastic functional differential equations. Substantially, these are stochastic differential equations with coefficients depending on the past history of the process itself. Such coefficients are hence defined on a functional space. Models with memory appear in many applications ranging from biology to finance. Here we consider the results of some evaluations based on these models (e.g. the prices of some financial products) and the risks conne...
On weak solutions of stochastic differential equations II
Czech Academy of Sciences Publication Activity Database
Hofmanová, Martina; Seidler, Jan
2013-01-01
Roč. 31, č. 4 (2013), s. 663-670 ISSN 0736-2994 R&D Projects: GA ČR GAP201/10/0752 Institutional support: RVO:67985556 Keywords : fractional integrals * stochastic differential equations * weak solutions Subject RIV: BA - General Mathematics Impact factor: 0.664, year: 2013 http://library.utia.cas.cz/separaty/2013/SI/hofmanova-on weak solutions of stochastic differential equations II.pdf
Comment on "Numerical methods for stochastic differential equations".
Burrage, Kevin; Burrage, Pamela; Higham, Desmond J; Kloeden, Peter E; Platen, Eckhard
2006-12-01
Wilkie [Phys. Rev. E 70, 017701 (2004)] used a heuristic approach to derive Runge-Kutta-based numerical methods for stochastic differential equations based on methods used for solving ordinary differential equations. The aim was to follow solution paths with high order. We point out that this approach is invalid in the general case and does not lead to high order methods. We warn readers against the inappropriate use of deterministic calculus in a stochastic setting.
Introduction to stochastic analysis integrals and differential equations
Mackevicius, Vigirdas
2013-01-01
This is an introduction to stochastic integration and stochastic differential equations written in an understandable way for a wide audience, from students of mathematics to practitioners in biology, chemistry, physics, and finances. The presentation is based on the naïve stochastic integration, rather than on abstract theories of measure and stochastic processes. The proofs are rather simple for practitioners and, at the same time, rather rigorous for mathematicians. Detailed application examples in natural sciences and finance are presented. Much attention is paid to simulation diffusion pro
Punshon-Smith, Samuel; Smith, Scott
2018-02-01
This article studies the Cauchy problem for the Boltzmann equation with stochastic kinetic transport. Under a cut-off assumption on the collision kernel and a coloring hypothesis for the noise coefficients, we prove the global existence of renormalized (in the sense of DiPerna/Lions) martingale solutions to the Boltzmann equation for large initial data with finite mass, energy, and entropy. Our analysis includes a detailed study of weak martingale solutions to a class of linear stochastic kinetic equations. This study includes a criterion for renormalization, the weak closedness of the solution set, and tightness of velocity averages in {{L}1}.
Effective action for stochastic partial differential equations.
Hochberg, D; Molina-París, C; Pérez-Mercader, J; Visser, M
1999-12-01
Stochastic partial differential equations (SPDEs) are the basic tool for modeling systems where noise is important. SPDEs are used for models of turbulence, pattern formation, and the structural development of the universe itself. It is reasonably well known that certain SPDEs can be manipulated to be equivalent to (nonquantum) field theories that nevertheless exhibit deep and important relationships with quantum field theory. In this paper we systematically extend these ideas: We set up a functional integral formalism and demonstrate how to extract all the one-loop physics for an arbitrary SPDE subject to arbitrary Gaussian noise. It is extremely important to realize that Gaussian noise does not imply that the field variables undergo Gaussian fluctuations, and that these nonquantum field theories are fully interacting. The limitation to one loop is not as serious as might be supposed: Experience with quantum field theories (QFTs) has taught us that one-loop physics is often quite adequate to give a good description of the salient issues. The limitation to one loop does, however, offer marked technical advantages: Because at one loop almost any field theory can be rendered finite using zeta function technology, we can sidestep the complications inherent in the Martin-Siggia-Rose formalism (the SPDE analog of the Becchi-Rouet-Stora-Tyutin formalism used in QFT) and instead focus attention on a minimalist approach that uses only the physical fields (this "direct approach" is the SPDE analog of canonical quantization using physical fields). After setting up the general formalism for the characteristic functional (partition function), we show how to define the effective action to all loops, and then focus on the one-loop effective action and its specialization to constant fields: the effective potential. The physical interpretation of the effective action and effective potential for SPDEs is addressed and we show that key features carry over from QFT to the case of
Exponential Stability of Stochastic Differential Equation with Mixed Delay
Directory of Open Access Journals (Sweden)
Wenli Zhu
2014-01-01
Full Text Available This paper focuses on a class of stochastic differential equations with mixed delay based on Lyapunov stability theory, Itô formula, stochastic analysis, and inequality technique. A sufficient condition for existence and uniqueness of the adapted solution to such systems is established by employing fixed point theorem. Some sufficient conditions of exponential stability and corollaries for such systems are obtained by using Lyapunov function. By utilizing Doob’s martingale inequality and Borel-Cantelli lemma, it is shown that the exponentially stable in the mean square of such systems implies the almost surely exponentially stable. In particular, our theoretical results show that if stochastic differential equation is exponentially stable and the time delay is sufficiently small, then the corresponding stochastic differential equation with mixed delay will remain exponentially stable. Moreover, time delay upper limit is solved by using our theoretical results when the system is exponentially stable, and they are more easily verified and applied in practice.
Stability theorems for stochastic differential equations driven by G-Brownian motion
Zhang, Defei
2011-01-01
In this paper, stability theorems for stochastic differential equations and backward stochastic differential equations driven by G-Brownian motion are obtained. We show the existence and uniqueness of solutions to forward-backward stochastic differential equations driven by G-Brownian motion. Stability theorem for forward-backward stochastic differential equations driven by G-Brownian motion is also presented.
Feynman path integral related to stochastic schroedinger equation
International Nuclear Information System (INIS)
Belavkin, V.P.; Smolyanov, O.G.
1998-01-01
The derivation of the Schroedinger equation describing the continuous measurement process is presented. The representation of the solution of the stochastic Schroedinger equation for continuous measurements is obtained by means of the Feynman path integral. The connection with the heuristic approach to the description of continuous measurements is considered. The connection with the Senon paradox is established [ru
Directory of Open Access Journals (Sweden)
M. M. Saleh
2007-01-01
Full Text Available The stochastic finite element method (SFEM is employed for solving stochastic one-dimension time-dependent differential equations with random coefficients. SFEM is used to have a fixed form of linear algebraic equations for polynomial chaos coefficients of the solution process. Four fixed forms are obtained in the cases of stochastic heat equation with stochastic heat capacity or heat conductivity coefficients and stochastic wave equation with stochastic mass density or elastic modulus coefficients. The relation between the exact deterministic solution and the mean of solution process is numerically studied.
Stochastic analysis of complex reaction networks using binomial moment equations.
Barzel, Baruch; Biham, Ofer
2012-09-01
The stochastic analysis of complex reaction networks is a difficult problem because the number of microscopic states in such systems increases exponentially with the number of reactive species. Direct integration of the master equation is thus infeasible and is most often replaced by Monte Carlo simulations. While Monte Carlo simulations are a highly effective tool, equation-based formulations are more amenable to analytical treatment and may provide deeper insight into the dynamics of the network. Here, we present a highly efficient equation-based method for the analysis of stochastic reaction networks. The method is based on the recently introduced binomial moment equations [Barzel and Biham, Phys. Rev. Lett. 106, 150602 (2011)]. The binomial moments are linear combinations of the ordinary moments of the probability distribution function of the population sizes of the interacting species. They capture the essential combinatorics of the reaction processes reflecting their stoichiometric structure. This leads to a simple and transparent form of the equations, and allows a highly efficient and surprisingly simple truncation scheme. Unlike ordinary moment equations, in which the inclusion of high order moments is prohibitively complicated, the binomial moment equations can be easily constructed up to any desired order. The result is a set of equations that enables the stochastic analysis of complex reaction networks under a broad range of conditions. The number of equations is dramatically reduced from the exponential proliferation of the master equation to a polynomial (and often quadratic) dependence on the number of reactive species in the binomial moment equations. The aim of this paper is twofold: to present a complete derivation of the binomial moment equations; to demonstrate the applicability of the moment equations for a representative set of example networks, in which stochastic effects play an important role.
Stochastic differential equations and turbulent dispersion
Durbin, P. A.
1983-01-01
Aspects of the theory of continuous stochastic processes that seem to contribute to an understanding of turbulent dispersion are introduced and the theory and philosophy of modelling turbulent transport is emphasized. Examples of eddy diffusion examined include shear dispersion, the surface layer, and channel flow. Modeling dispersion with finite-time scale is considered including the Langevin model for homogeneous turbulence, dispersion in nonhomogeneous turbulence, and the asymptotic behavior of the Langevin model for nonhomogeneous turbulence.
Spontaneous Stochasticity and Anomalous Dissipation for Burgers Equation
Eyink, Gregory L.; Drivas, Theodore D.
2015-01-01
We develop a Lagrangian approach to conservation-law anomalies in weak solutions of inviscid Burgers equation, motivated by previous work on the Kraichnan model of turbulent scalar advection. We show that the entropy solutions of Burgers possess Markov stochastic processes of (generalized) Lagrangian trajectories backward in time for which the Burgers velocity is a backward martingale. This property is shown to guarantee dissipativity of conservation-law anomalies for general convex functions of the velocity. The backward stochastic Burgers flows with these properties are not unique, however. We construct infinitely many such stochastic flows, both by a geometric construction and by the zero-noise limit of the Constantin-Iyer stochastic representation of viscous Burgers solutions. The latter proof yields the spontaneous stochasticity of Lagrangian trajectories backward in time for Burgers, at unit Prandtl number. It is conjectured that existence of a backward stochastic flow with the velocity as martingale is an admissibility condition which selects the unique entropy solution for Burgers. We also study linear transport of passive densities and scalars by inviscid Burgers flows. We show that shock solutions of Burgers exhibit spontaneous stochasticity backward in time for all finite Prandtl numbers, implying conservation-law anomalies for linear transport. We discuss the relation of our results for Burgers with incompressible Navier-Stokes turbulence, especially Lagrangian admissibility conditions for Euler solutions and the relation between turbulent cascade directions and time-asymmetry of Lagrangian stochasticity.
Necessary Conditions for Optimal Control of Stochastic Evolution Equations in Hilbert Spaces
Energy Technology Data Exchange (ETDEWEB)
Al-Hussein, Abdul Rahman, E-mail: alhusseinqu@hotmail.com [Qassim University, Department of Mathematics, College of Science (Saudi Arabia)
2011-06-15
We consider a nonlinear stochastic optimal control problem associated with a stochastic evolution equation. This equation is driven by a continuous martingale in a separable Hilbert space and an unbounded time-dependent linear operator.We derive a stochastic maximum principle for this optimal control problem. Our results are achieved by using the adjoint backward stochastic partial differential equation.
Roerdink, J.B.T.M.
1982-01-01
It is shown that the cumulant expansion for linear stochastic differential equations, hitherto used to compute one-time averages of the solution process, is also capable of yielding the two-time correlation and probability density functions. The general case with a coefficient matrix, an
International Nuclear Information System (INIS)
Ge, Gen; Li, ZePeng
2016-01-01
A modified stochastic averaging method on single-degree-of-freedom (SDOF) oscillators under white noise excitations with strongly nonlinearity was proposed. Considering the existing approach dealing with strongly nonlinear SDOFs derived by Zhu and Huang [14, 15] is quite time consuming in calculating the drift coefficient and diffusion coefficients and the expressions of them are considerable long, the so-called He's energy balance method was applied to overcome the minor defect of the Zhu and Huang's method. The modified method can offer more concise approximate expressions of the drift and diffusion coefficients without weakening the accuracy of predicting the responses of the systems too much by giving an averaged frequency beforehand. Three examples, a cubic and quadratic nonlinearity coexisting oscillator, a quadratic nonlinear oscillator under external white noise excitations and an externally excited Duffing–Rayleigh oscillator, were given to illustrate the approach we proposed. The three examples were excited by the Gaussian white noise and the Gaussian colored noise separately. The stationary responses of probability density of amplitudes and energy, together with joint probability density of displacement and velocity are studied to verify the presented approach. The reliability of the systems were also investigated to offer further support. Digital simulations were carried out and the output of that are coincide with the theoretical approximations well.
pth Moment Exponential Stability of Nonlinear Hybrid Stochastic Heat Equations
Directory of Open Access Journals (Sweden)
Xuetao Yang
2014-01-01
Full Text Available We are concerned with the exponential stability problem of a class of nonlinear hybrid stochastic heat equations (known as stochastic heat equations with Markovian switching in an infinite state space. The fixed point theory is utilized to discuss the existence, uniqueness, and pth moment exponential stability of the mild solution. Moreover, we also acquire the Lyapunov exponents by combining the fixed point theory and the Gronwall inequality. At last, two examples are provided to verify the effectiveness of our obtained results.
Multiple-scale stochastic processes: Decimation, averaging and beyond
Energy Technology Data Exchange (ETDEWEB)
Bo, Stefano, E-mail: stefano.bo@nordita.org [Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, SE-106 91 Stockholm (Sweden); Celani, Antonio [Quantitative Life Sciences, The Abdus Salam International Centre for Theoretical Physics (ICTP), Strada Costiera 11, I-34151 - Trieste (Italy)
2017-02-07
The recent experimental progresses in handling microscopic systems have allowed to probe them at levels where fluctuations are prominent, calling for stochastic modeling in a large number of physical, chemical and biological phenomena. This has provided fruitful applications for established stochastic methods and motivated further developments. These systems often involve processes taking place on widely separated time scales. For an efficient modeling one usually focuses on the slower degrees of freedom and it is of great importance to accurately eliminate the fast variables in a controlled fashion, carefully accounting for their net effect on the slower dynamics. This procedure in general requires to perform two different operations: decimation and coarse-graining. We introduce the asymptotic methods that form the basis of this procedure and discuss their application to a series of physical, biological and chemical examples. We then turn our attention to functionals of the stochastic trajectories such as residence times, counting statistics, fluxes, entropy production, etc. which have been increasingly studied in recent years. For such functionals, the elimination of the fast degrees of freedom can present additional difficulties and naive procedures can lead to blatantly inconsistent results. Homogenization techniques for functionals are less covered in the literature and we will pedagogically present them here, as natural extensions of the ones employed for the trajectories. We will also discuss recent applications of these techniques to the thermodynamics of small systems and their interpretation in terms of information-theoretic concepts.
Directory of Open Access Journals (Sweden)
Hui Min
2014-01-01
Full Text Available We discuss a new type of fully coupled forward-backward stochastic differential equations (FBSDEs whose coefficients depend on the states of the solution processes as well as their expected values, and we call them fully coupled mean-field forward-backward stochastic differential equations (mean-field FBSDEs. We first prove the existence and the uniqueness theorem of such mean-field FBSDEs under some certain monotonicity conditions and show the continuity property of the solutions with respect to the parameters. Then we discuss the stochastic optimal control problems of mean-field FBSDEs. The stochastic maximum principles are derived and the related mean-field linear quadratic optimal control problems are also discussed.
2–stage stochastic Runge–Kutta for stochastic delay differential equations
Energy Technology Data Exchange (ETDEWEB)
Rosli, Norhayati; Jusoh Awang, Rahimah [Faculty of Industrial Science and Technology, Universiti Malaysia Pahang, Lebuhraya Tun Razak, 26300, Gambang, Pahang (Malaysia); Bahar, Arifah; Yeak, S. H. [Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 Johor Bahru, Johor (Malaysia)
2015-05-15
This paper proposes a newly developed one-step derivative-free method, that is 2-stage stochastic Runge-Kutta (SRK2) to approximate the solution of stochastic delay differential equations (SDDEs) with a constant time lag, r > 0. General formulation of stochastic Runge-Kutta for SDDEs is introduced and Stratonovich Taylor series expansion for numerical solution of SRK2 is presented. Local truncation error of SRK2 is measured by comparing the Stratonovich Taylor expansion of the exact solution with the computed solution. Numerical experiment is performed to assure the validity of the method in simulating the strong solution of SDDEs.
A Simple Stochastic Differential Equation with Discontinuous Drift
DEFF Research Database (Denmark)
Simonsen, Maria; Leth, John-Josef; Schiøler, Henrik
2013-01-01
In this paper we study solutions to stochastic differential equations (SDEs) with discontinuous drift. We apply two approaches: The Euler-Maruyama method and the Fokker-Planck equation and show that a candidate density function based on the Euler-Maruyama method approximates a candidate density...... function based on the stationary Fokker-Planck equation. Furthermore, we introduce a smooth function which approximates the discontinuous drift and apply the Euler-Maruyama method and the Fokker-Planck equation with this input. The point of departure for this work is a particular SDE with discontinuous...
Master equations and the theory of stochastic path integrals
Weber, Markus F.; Frey, Erwin
2017-04-01
This review provides a pedagogic and self-contained introduction to master equations and to their representation by path integrals. Since the 1930s, master equations have served as a fundamental tool to understand the role of fluctuations in complex biological, chemical, and physical systems. Despite their simple appearance, analyses of master equations most often rely on low-noise approximations such as the Kramers-Moyal or the system size expansion, or require ad-hoc closure schemes for the derivation of low-order moment equations. We focus on numerical and analytical methods going beyond the low-noise limit and provide a unified framework for the study of master equations. After deriving the forward and backward master equations from the Chapman-Kolmogorov equation, we show how the two master equations can be cast into either of four linear partial differential equations (PDEs). Three of these PDEs are discussed in detail. The first PDE governs the time evolution of a generalized probability generating function whose basis depends on the stochastic process under consideration. Spectral methods, WKB approximations, and a variational approach have been proposed for the analysis of the PDE. The second PDE is novel and is obeyed by a distribution that is marginalized over an initial state. It proves useful for the computation of mean extinction times. The third PDE describes the time evolution of a ‘generating functional’, which generalizes the so-called Poisson representation. Subsequently, the solutions of the PDEs are expressed in terms of two path integrals: a ‘forward’ and a ‘backward’ path integral. Combined with inverse transformations, one obtains two distinct path integral representations of the conditional probability distribution solving the master equations. We exemplify both path integrals in analysing elementary chemical reactions. Moreover, we show how a well-known path integral representation of averaged observables can be recovered from
Master equations and the theory of stochastic path integrals.
Weber, Markus F; Frey, Erwin
2017-04-01
This review provides a pedagogic and self-contained introduction to master equations and to their representation by path integrals. Since the 1930s, master equations have served as a fundamental tool to understand the role of fluctuations in complex biological, chemical, and physical systems. Despite their simple appearance, analyses of master equations most often rely on low-noise approximations such as the Kramers-Moyal or the system size expansion, or require ad-hoc closure schemes for the derivation of low-order moment equations. We focus on numerical and analytical methods going beyond the low-noise limit and provide a unified framework for the study of master equations. After deriving the forward and backward master equations from the Chapman-Kolmogorov equation, we show how the two master equations can be cast into either of four linear partial differential equations (PDEs). Three of these PDEs are discussed in detail. The first PDE governs the time evolution of a generalized probability generating function whose basis depends on the stochastic process under consideration. Spectral methods, WKB approximations, and a variational approach have been proposed for the analysis of the PDE. The second PDE is novel and is obeyed by a distribution that is marginalized over an initial state. It proves useful for the computation of mean extinction times. The third PDE describes the time evolution of a 'generating functional', which generalizes the so-called Poisson representation. Subsequently, the solutions of the PDEs are expressed in terms of two path integrals: a 'forward' and a 'backward' path integral. Combined with inverse transformations, one obtains two distinct path integral representations of the conditional probability distribution solving the master equations. We exemplify both path integrals in analysing elementary chemical reactions. Moreover, we show how a well-known path integral representation of averaged observables can be recovered from them. Upon
Stochastic Simulation of Hourly Average Wind Speed in Umudike ...
African Journals Online (AJOL)
Ten years of hourly average wind speed data were used to build a seasonal autoregressive integrated moving average (SARIMA) model. The model was used to simulate hourly average wind speed and recommend possible uses at Umudike, South eastern Nigeria. Results showed that the simulated wind behaviour was ...
LINEAR MULTIFRACTIONAL STOCHASTIC VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS
Nguyen, Tien Dung
2013-01-01
In this paper we prove the variation of parameters formula for linear Volterra integro-differential equations driven by multifractional Brownian motion. To do this, an approximate result for the Stratonovich stochastic integral with respect to the multifractional Brownian motion is given. Based on our obtained results we study almost surely exponentially convergence of the solution. Also, the existence and uniqueness of the solution of a multifractional Volterra integro-differential equation ...
High Weak Order Methods for Stochastic Differential Equations Based on Modified Equations
Abdulle, Assyr
2012-01-01
© 2012 Society for Industrial and Applied Mathematics. Inspired by recent advances in the theory of modified differential equations, we propose a new methodology for constructing numerical integrators with high weak order for the time integration of stochastic differential equations. This approach is illustrated with the constructions of new methods of weak order two, in particular, semi-implicit integrators well suited for stiff (meansquare stable) stochastic problems, and implicit integrators that exactly conserve all quadratic first integrals of a stochastic dynamical system. Numerical examples confirm the theoretical results and show the versatility of our methodology.
Derivation of Stochastic Equations for Computational Uncertainties ...
African Journals Online (AJOL)
ADOWIE PERE
physical parameters using the relative standard deviations techniques (σ). These equations also known as ... define the maximum level of uncertainty that can be tolerated in any independent variable if the maximum uncertainty to be ... standard deviations, partial derivatives, degree of accuracy, empirical models. LIST OF ...
Reflected backward stochastic differential equations in an orthant
Indian Academy of Sciences (India)
R. Narasimhan (Krishtel eMaging) 1461 1996 Oct 15 13:05:22
differential games and economics. Recently backward stochastic differential equations with reflecting barrier have been studied by El Karoui et al [5] and Cvitanic and Karatzas. [1] in the one-dimensional case; and by Gegout-Petit and Pardoux [7] in a convex domain in higher dimensions; these works concern the case of ...
Efficient Numerical Methods for Stochastic Differential Equations in Computational Finance
Happola, Juho
2017-09-19
Stochastic Differential Equations (SDE) offer a rich framework to model the probabilistic evolution of the state of a system. Numerical approximation methods are typically needed in evaluating relevant Quantities of Interest arising from such models. In this dissertation, we present novel effective methods for evaluating Quantities of Interest relevant to computational finance when the state of the system is described by an SDE.
A class of degenerate stochastic differential equations with non ...
Indian Academy of Sciences (India)
Introduction. In this article we consider (possibly degenerate) stochastic differential equations (SDEs) with non-Lipschitz coefficients. If the coefficients are Lipschitz, we can prove the existence of a unique strong solution (see [9]). But uniqueness fails in the case of non-Lipschitz coefficients. The literature on this topic is not ...
Unigueness for stochastic non-linear wave equations
Czech Academy of Sciences Publication Activity Database
Ondreját, Martin
2007-01-01
Roč. 67, č. 12 (2007), s. 3287-3310 ISSN 0362-546X R&D Projects: GA ČR GA201/04/0750 Institutional research plan: CEZ:AV0Z10190503 Keywords : stochastic wave equation * uniqueness Subject RIV: BA - General Mathematics Impact factor: 1.097, year: 2007
Mild Solutions of Neutral Stochastic Partial Functional Differential Equations
Directory of Open Access Journals (Sweden)
T. E. Govindan
2011-01-01
Full Text Available This paper studies the existence and uniqueness of a mild solution for a neutral stochastic partial functional differential equation using a local Lipschitz condition. When the neutral term is zero and even in the deterministic special case, the result obtained here appears to be new. An example is included to illustrate the theory.
Stability of Solutions to Semilinear Stochastic Evolution Equations
Czech Academy of Sciences Publication Activity Database
Leha, G.; Maslowski, Bohdan; Ritter, G.
1999-01-01
Roč. 17, č. 6 (1999), s. 1009-1051 ISSN 0736-2994 R&D Projects: GA ČR GA201/95/0629 Institutional research plan: CEZ:AV0Z1019905; CEZ:AV0Z1019905 Keywords : stochastic evolution equations * Lyapunov stability * forward inequality Subject RIV: BA - General Mathematics Impact factor: 0.263, year: 1999
A stochastic differential equation framework for the turbulent velocity field
DEFF Research Database (Denmark)
Barndorff-Nielsen, Ole Eiler; Schmiegel, Jürgen
We discuss a stochastic differential equation, as a modelling framework for the turbulent velocity field, that is capable of capturing basic stylized facts of the statistics of velocity increments. In particular, we focus on the evolution of the probability density of velocity increments...
Improved stochastic approximation methods for discretized parabolic partial differential equations
Guiaş, Flavius
2016-12-01
We present improvements of the stochastic direct simulation method, a known numerical scheme based on Markov jump processes which is used for approximating solutions of ordinary differential equations. This scheme is suited especially for spatial discretizations of evolution partial differential equations (PDEs). By exploiting the full path simulation of the stochastic method, we use this first approximation as a predictor and construct improved approximations by Picard iterations, Runge-Kutta steps, or a combination. This has as consequence an increased order of convergence. We illustrate the features of the improved method at a standard benchmark problem, a reaction-diffusion equation modeling a combustion process in one space dimension (1D) and two space dimensions (2D).
Application of Stochastic Partial Differential Equations to Reservoir Property Modelling
Potsepaev, R.
2010-09-06
Existing algorithms of geostatistics for stochastic modelling of reservoir parameters require a mapping (the \\'uvt-transform\\') into the parametric space and reconstruction of a stratigraphic co-ordinate system. The parametric space can be considered to represent a pre-deformed and pre-faulted depositional environment. Existing approximations of this mapping in many cases cause significant distortions to the correlation distances. In this work we propose a coordinate free approach for modelling stochastic textures through the application of stochastic partial differential equations. By avoiding the construction of a uvt-transform and stratigraphic coordinates, one can generate realizations directly in the physical space in the presence of deformations and faults. In particular the solution of the modified Helmholtz equation driven by Gaussian white noise is a zero mean Gaussian stationary random field with exponential correlation function (in 3-D). This equation can be used to generate realizations in parametric space. In order to sample in physical space we introduce a stochastic elliptic PDE with tensor coefficients, where the tensor is related to correlation anisotropy and its variation is physical space.
Indirect Inference for Stochastic Differential Equations Based on Moment Expansions
Ballesio, Marco
2016-01-06
We provide an indirect inference method to estimate the parameters of timehomogeneous scalar diffusion and jump diffusion processes. We obtain a system of ODEs that approximate the time evolution of the first two moments of the process by the approximation of the stochastic model applying a second order Taylor expansion of the SDE s infinitesimal generator in the Dynkin s formula. This method allows a simple and efficient procedure to infer the parameters of such stochastic processes given the data by the maximization of the likelihood of an approximating Gaussian process described by the two moments equations. Finally, we perform numerical experiments for two datasets arising from organic and inorganic fouling deposition phenomena.
Stochastic Calculus and Differential Equations for Physics and Finance
McCauley, Joseph L.
2013-02-01
1. Random variables and probability distributions; 2. Martingales, Markov, and nonstationarity; 3. Stochastic calculus; 4. Ito processes and Fokker-Planck equations; 5. Selfsimilar Ito processes; 6. Fractional Brownian motion; 7. Kolmogorov's PDEs and Chapman-Kolmogorov; 8. Non Markov Ito processes; 9. Black-Scholes, martingales, and Feynman-Katz; 10. Stochastic calculus with martingales; 11. Statistical physics and finance, a brief history of both; 12. Introduction to new financial economics; 13. Statistical ensembles and time series analysis; 14. Econometrics; 15. Semimartingales; References; Index.
Reflected stochastic differential equation models for constrained animal movement
Hanks, Ephraim M.; Johnson, Devin S.; Hooten, Mevin B.
2017-01-01
Movement for many animal species is constrained in space by barriers such as rivers, shorelines, or impassable cliffs. We develop an approach for modeling animal movement constrained in space by considering a class of constrained stochastic processes, reflected stochastic differential equations. Our approach generalizes existing methods for modeling unconstrained animal movement. We present methods for simulation and inference based on augmenting the constrained movement path with a latent unconstrained path and illustrate this augmentation with a simulation example and an analysis of telemetry data from a Steller sea lion (Eumatopias jubatus) in southeast Alaska.
Ren, Jiagang; Wu, Jing; Zhang, Hua
2015-01-01
In this paper, we prove a large deviation principle of Freidlin-Wentzell's type for the multivalued stochastic differential equations. As an application, we derive a functional iterated logarithm law for the solutions of multivalued stochastic differential equations.
Propagation of ultra-short solitons in stochastic Maxwell's equations
Energy Technology Data Exchange (ETDEWEB)
Kurt, Levent, E-mail: LKurt@gc.cuny.edu [Department of Science, Borough of Manhattan Community College, City University of New York, New York, New York 10007 (United States); Schäfer, Tobias [Department of Mathematics, College of Staten Island, City University of New York, Staten Island, New York 10314 (United States)
2014-01-15
We study the propagation of ultra-short short solitons in a cubic nonlinear medium modeled by nonlinear Maxwell's equations with stochastic variations of media. We consider three cases: variations of (a) the dispersion, (b) the phase velocity, (c) the nonlinear coefficient. Using a modified multi-scale expansion for stochastic systems, we derive new stochastic generalizations of the short pulse equation that approximate the solutions of stochastic nonlinear Maxwell's equations. Numerical simulations show that soliton solutions of the short pulse equation propagate stably in stochastic nonlinear Maxwell's equations and that the generalized stochastic short pulse equations approximate the solutions to the stochastic Maxwell's equations over the distances under consideration. This holds for both a pathwise comparison of the stochastic equations as well as for a comparison of the resulting probability densities.
Stochastic Evolution Equations Driven by Fractional Noises
2016-11-28
PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 5c. PROGRAM ELEMENT NUMBER 5b. GRANT NUMBER 5a. CONTRACT NUMBER Form Approved OMB NO. 0704...rate of convergence to zero or the error and the limit in distribution of the error fluctuations. We have studied time discrete numerical schemes...to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data
On the Existence and the Applications of Modified Equations for Stochastic Differential Equations
Zygalakis, K. C.
2011-01-01
In this paper we describe a general framework for deriving modified equations for stochastic differential equations (SDEs) with respect to weak convergence. Modified equations are derived for a variety of numerical methods, such as the Euler or the Milstein method. Existence of higher order modified equations is also discussed. In the case of linear SDEs, using the Gaussianity of the underlying solutions, we derive an SDE which the numerical method solves exactly in the weak sense. Applications of modified equations in the numerical study of Langevin equations is also discussed. © 2011 Society for Industrial and Applied Mathematics.
Numerical method for solving stochastic differential equations with dichotomous noise.
Kim, Changho; Lee, Eok Kyun; Talkner, Peter
2006-02-01
We propose a numerical method for solving stochastic differential equations with dichotomous Markov noise. The numerical scheme is formulated such that (i) the stochastic formula used follows the Stratonovich-Taylor form over the entire range of noise correlation times, including the Gaussian white noise limit; and (ii) the method is readily applicable to dynamical systems driven by arbitrary types of noise, provided there exists a way to describe the random increment of the stochastic process expressed in the Stratonovich-Taylor form. We further propose a simplified Taylor scheme that significantly reduces the computation time, while still satisfying the moment properties up to the required order. The accuracies and efficiencies of the proposed algorithms are validated by applying the schemes to two prototypical model systems that possess analytical solutions.
Stochastic integration in Banach spaces and applications to parabolic evolution equations
Veraar, M.C.
2006-01-01
Stochastic partial differential equations (SPDEs) of evolution type are usually modelled as ordinary stochastic differential equations (SDEs) in an infinite-dimensional state space. In many examples such as the stochastic heat and wave equation, this viewpoint may lead to existence and uniqueness
Excitability in a stochastic differential equation model for calcium puffs.
Rüdiger, S
2014-06-01
Calcium dynamics are essential to a multitude of cellular processes. For many cell types, localized discharges of calcium through small clusters of intracellular channels are building blocks for all spatially extended calcium signals. Because of the large noise amplitude, the validity of noise-approximating model equations for this system has been questioned. Here we revisit the master equations for local calcium release, examine the multiple scales of calcium concentrations in the cluster domain, and derive adapted stochastic differential equations. We show by comparison of discrete and continuous trajectories that the Langevin equations can be made consistent with the master equations even for very small channel numbers. In its deterministic limit, the model reveals that excitability, a dynamical phenomenon observed in many natural systems, is at the core of calcium puffs. The model also predicts a bifurcation from transient to sustained release which may link local and global calcium signals in cells.
Modeling ion channel dynamics through reflected stochastic differential equations.
Dangerfield, Ciara E; Kay, David; Burrage, Kevin
2012-05-01
Ion channels are membrane proteins that open and close at random and play a vital role in the electrical dynamics of excitable cells. The stochastic nature of the conformational changes these proteins undergo can be significant, however current stochastic modeling methodologies limit the ability to study such systems. Discrete-state Markov chain models are seen as the "gold standard," but are computationally intensive, restricting investigation of stochastic effects to the single-cell level. Continuous stochastic methods that use stochastic differential equations (SDEs) to model the system are more efficient but can lead to simulations that have no biological meaning. In this paper we show that modeling the behavior of ion channel dynamics by a reflected SDE ensures biologically realistic simulations, and we argue that this model follows from the continuous approximation of the discrete-state Markov chain model. Open channel and action potential statistics from simulations of ion channel dynamics using the reflected SDE are compared with those of a discrete-state Markov chain method. Results show that the reflected SDE simulations are in good agreement with the discrete-state approach. The reflected SDE model therefore provides a computationally efficient method to simulate ion channel dynamics while preserving the distributional properties of the discrete-state Markov chain model and also ensuring biologically realistic solutions. This framework could easily be extended to other biochemical reaction networks.
Algorithm refinement for stochastic partial differential equations I. linear diffusion
Alexander, F J; Tartakovsky, D M
2002-01-01
A hybrid particle/continuum algorithm is formulated for Fickian diffusion in the fluctuating hydrodynamic limit. The particles are taken as independent random walkers; the fluctuating diffusion equation is solved by finite differences with deterministic and white-noise fluxes. At the interface between the particle and continuum computations the coupling is by flux matching, giving exact mass conservation. This methodology is an extension of Adaptive Mesh and Algorithm Refinement to stochastic partial differential equations. Results from a variety of numerical experiments are presented for both steady and time-dependent scenarios. In all cases the mean and variance of density are captured correctly by the stochastic hybrid algorithm. For a nonstochastic version (i.e., using only deterministic continuum fluxes) the mean density is correct, but the variance is reduced except in particle regions away from the interface. Extensions of the methodology to fluid mechanics applications are discussed.
A stochastic differential equation model for transcriptional regulatory networks
Directory of Open Access Journals (Sweden)
Quirk Michelle D
2007-05-01
Full Text Available Abstract Background This work explores the quantitative characteristics of the local transcriptional regulatory network based on the availability of time dependent gene expression data sets. The dynamics of the gene expression level are fitted via a stochastic differential equation model, yielding a set of specific regulators and their contribution. Results We show that a beta sigmoid function that keeps track of temporal parameters is a novel prototype of a regulatory function, with the effect of improving the performance of the profile prediction. The stochastic differential equation model follows well the dynamic of the gene expression levels. Conclusion When adapted to biological hypotheses and combined with a promoter analysis, the method proposed here leads to improved models of the transcriptional regulatory networks.
Numerical Solution of Stochastic Nonlinear Fractional Differential Equations
El-Beltagy, Mohamed A.
2015-01-07
Using Wiener-Hermite expansion (WHE) technique in the solution of the stochastic partial differential equations (SPDEs) has the advantage of converting the problem to a system of deterministic equations that can be solved efficiently using the standard deterministic numerical methods [1]. WHE is the only known expansion that handles the white/colored noise exactly. This work introduces a numerical estimation of the stochastic response of the Duffing oscillator with fractional or variable order damping and driven by white noise. The WHE technique is integrated with the Grunwald-Letnikov approximation in case of fractional order and with Coimbra approximation in case of variable-order damping. The numerical solver was tested with the analytic solution and with Monte-Carlo simulations. The developed mixed technique was shown to be efficient in simulating SPDEs.
Probabilistic Forecasts of Solar Irradiance by Stochastic Differential Equations
DEFF Research Database (Denmark)
Iversen, Jan Emil Banning; Morales González, Juan Miguel; Møller, Jan Kloppenborg
2014-01-01
Probabilistic forecasts of renewable energy production provide users with valuable information about the uncertainty associated with the expected generation. Current state-of-the-art forecasts for solar irradiance have focused on producing reliable point forecasts. The additional information...... included in probabilistic forecasts may be paramount for decision makers to efficiently make use of this uncertain and variable generation. In this paper, a stochastic differential equation framework for modeling the uncertainty associated with the solar irradiance point forecast is proposed. This modeling...... approach allows for characterizing both the interdependence structure of prediction errors of short-term solar irradiance and their predictive distribution. Three different stochastic differential equation models are first fitted to a training data set and subsequently evaluated on a one-year test set...
On Almost Automorphic Mild Solutions for Nonautonomous Stochastic Evolution Equations
Directory of Open Access Journals (Sweden)
Jing Cui
2012-01-01
Full Text Available We consider a class of nonautonomous stochastic evolution equations in real separable Hilbert spaces. We establish a new composition theorem for square-mean almost automorphic functions under non-Lipschitz conditions. We apply this new composition theorem as well as intermediate space techniques, Krasnoselskii fixed point theorem, and Banach fixed point theorem to investigate the existence of square-mean almost automorphic mild solutions. Some known results are generalized and improved.
Strong solutions of semilinear stochastic partial differential equations
Czech Academy of Sciences Publication Activity Database
Hofmanová, Martina
2013-01-01
Roč. 20, č. 3 (2013), s. 757-778 ISSN 1021-9722 R&D Projects: GA ČR GAP201/10/0752 Institutional support: RVO:67985556 Keywords : stochastic partial differential equations * strongly elliptic differential operator * strongly continuous semigroup Subject RIV: BA - General Mathematics Impact factor: 0.971, year: 2013 http://library.utia.cas.cz/separaty/2013/SI/hofmanova-0393085.pdf
Stochastic wave equation with critical nonlinearities: Temporal regularity and uniqueness
Czech Academy of Sciences Publication Activity Database
Ondreját, Martin
2010-01-01
Roč. 248, č. 7 (2010), s. 1579-1602 ISSN 0022-0396 R&D Projects: GA ČR GA201/07/0237 Institutional research plan: CEZ:AV0Z10750506 Keywords : Stochastic wave equation * Spatially homogeneous Wiener process * Critical growth Subject RIV: BA - General Mathematics Impact factor: 1.261, year: 2010 http://library.utia.cas.cz/separaty/2010/SI/ondrejat-0339543.pdf
Different seeds to solve the equations of stochastic point kinetics using the Euler-Maruyama method
International Nuclear Information System (INIS)
Suescun D, D.; Oviedo T, M.
2017-09-01
In this paper, a numerical study of stochastic differential equations that describe the kinetics in a nuclear reactor is presented. These equations, known as the stochastic equations of punctual kinetics they model temporal variations in neutron population density and concentrations of deferred neutron precursors. Because these equations are probabilistic in nature (since random oscillations in the neutrons and population of precursors were considered to be approximately normally distributed, and these equations also possess strong coupling and stiffness properties) the proposed method for the numerical simulations is the Euler-Maruyama scheme that provides very good approximations for calculating the neutron population and concentrations of deferred neutron precursors. The method proposed for this work was computationally tested for different seeds, initial conditions, experimental data and forms of reactivity for a group of precursors and then for six groups of deferred neutron precursors at each time step with 5000 Brownian movements per seed. In a paper reported in the literature, the Euler-Maruyama method was proposed, but there are many doubts about the reported values, in addition to not reporting the seed used, so in this work is expected to rectify the reported values. After taking the average of the different seeds used to generate the pseudo-random numbers the results provided by the Euler-Maruyama scheme will be compared in mean and standard deviation with other methods reported in the literature and results of the deterministic model of the equations of the punctual kinetics. This comparison confirms in particular that the Euler-Maruyama scheme is an efficient method to solve the equations of stochastic point kinetics but different from the values found and reported by another author. The Euler-Maruyama method is simple and easy to implement, provides acceptable results for neutron population density and concentration of deferred neutron precursors and
A stochastic Galerkin method for the Euler equations with Roe variable transformation
Pettersson, Per
2014-01-01
The Euler equations subject to uncertainty in the initial and boundary conditions are investigated via the stochastic Galerkin approach. We present a new fully intrusive method based on a variable transformation of the continuous equations. Roe variables are employed to get quadratic dependence in the flux function and a well-defined Roe average matrix that can be determined without matrix inversion.In previous formulations based on generalized polynomial chaos expansion of the physical variables, the need to introduce stochastic expansions of inverse quantities, or square roots of stochastic quantities of interest, adds to the number of possible different ways to approximate the original stochastic problem. We present a method where the square roots occur in the choice of variables, resulting in an unambiguous problem formulation.The Roe formulation saves computational cost compared to the formulation based on expansion of conservative variables. Moreover, the Roe formulation is more robust and can handle cases of supersonic flow, for which the conservative variable formulation fails to produce a bounded solution. For certain stochastic basis functions, the proposed method can be made more effective and well-conditioned. This leads to increased robustness for both choices of variables. We use a multi-wavelet basis that can be chosen to include a large number of resolution levels to handle more extreme cases (e.g. strong discontinuities) in a robust way. For smooth cases, the order of the polynomial representation can be increased for increased accuracy. © 2013 Elsevier Inc.
Homogenization of the stochastic Navier–Stokes equation with a stochastic slip boundary condition
Bessaih, Hakima
2015-11-02
The two-dimensional Navier–Stokes equation in a perforated domain with a dynamical slip boundary condition is considered. We assume that the dynamic is driven by a stochastic perturbation on the interior of the domain and another stochastic perturbation on the boundaries of the holes. We consider a scaling (ᵋ for the viscosity and 1 for the density) that will lead to a time-dependent limit problem. However, the noncritical scaling (ᵋ, β > 1) is considered in front of the nonlinear term. The homogenized system in the limit is obtained as a Darcy’s law with memory with two permeabilities and an extra term that is due to the stochastic perturbation on the boundary of the holes. The nonhomogeneity on the boundary contains a stochastic part that yields in the limit an additional term in the Darcy’s law. We use the two-scale convergence method after extending the solution with 0 inside the holes to pass to the limit. By Itô stochastic calculus, we get uniform estimates on the solution in appropriate spaces. Due to the stochastic integral, the pressure that appears in the variational formulation does not have enough regularity in time. This fact made us rely only on the variational formulation for the passage to the limit on the solution. We obtain a variational formulation for the limit that is solution of a Stokes system with two pressures. This two-scale limit gives rise to three cell problems, two of them give the permeabilities while the third one gives an extra term in the Darcy’s law due to the stochastic perturbation on the boundary of the holes.
Modelling biochemical reaction systems by stochastic differential equations with reflection.
Niu, Yuanling; Burrage, Kevin; Chen, Luonan
2016-05-07
In this paper, we gave a new framework for modelling and simulating biochemical reaction systems by stochastic differential equations with reflection not in a heuristic way but in a mathematical way. The model is computationally efficient compared with the discrete-state Markov chain approach, and it ensures that both analytic and numerical solutions remain in a biologically plausible region. Specifically, our model mathematically ensures that species numbers lie in the domain D, which is a physical constraint for biochemical reactions, in contrast to the previous models. The domain D is actually obtained according to the structure of the corresponding chemical Langevin equations, i.e., the boundary is inherent in the biochemical reaction system. A variant of projection method was employed to solve the reflected stochastic differential equation model, and it includes three simple steps, i.e., Euler-Maruyama method was applied to the equations first, and then check whether or not the point lies within the domain D, and if not perform an orthogonal projection. It is found that the projection onto the closure D¯ is the solution to a convex quadratic programming problem. Thus, existing methods for the convex quadratic programming problem can be employed for the orthogonal projection map. Numerical tests on several important problems in biological systems confirmed the efficiency and accuracy of this approach. Copyright © 2016 Elsevier Ltd. All rights reserved.
Stochastic partial differential equations a modeling, white noise functional approach
Holden, Helge; Ubøe, Jan; Zhang, Tusheng
1996-01-01
This book is based on research that, to a large extent, started around 1990, when a research project on fluid flow in stochastic reservoirs was initiated by a group including some of us with the support of VISTA, a research coopera tion between the Norwegian Academy of Science and Letters and Den norske stats oljeselskap A.S. (Statoil). The purpose of the project was to use stochastic partial differential equations (SPDEs) to describe the flow of fluid in a medium where some of the parameters, e.g., the permeability, were stochastic or "noisy". We soon realized that the theory of SPDEs at the time was insufficient to handle such equations. Therefore it became our aim to develop a new mathematically rigorous theory that satisfied the following conditions. 1) The theory should be physically meaningful and realistic, and the corre sponding solutions should make sense physically and should be useful in applications. 2) The theory should be general enough to handle many of the interesting SPDEs that occur in r...
Stochastic Differential Equation-Based Flexible Software Reliability Growth Model
Directory of Open Access Journals (Sweden)
P. K. Kapur
2009-01-01
Full Text Available Several software reliability growth models (SRGMs have been developed by software developers in tracking and measuring the growth of reliability. As the size of software system is large and the number of faults detected during the testing phase becomes large, so the change of the number of faults that are detected and removed through each debugging becomes sufficiently small compared with the initial fault content at the beginning of the testing phase. In such a situation, we can model the software fault detection process as a stochastic process with continuous state space. In this paper, we propose a new software reliability growth model based on Itô type of stochastic differential equation. We consider an SDE-based generalized Erlang model with logistic error detection function. The model is estimated and validated on real-life data sets cited in literature to show its flexibility. The proposed model integrated with the concept of stochastic differential equation performs comparatively better than the existing NHPP-based models.
DEFF Research Database (Denmark)
Chon, K H; Hoyer, D; Armoundas, A A
1999-01-01
In this study, we introduce a new approach for estimating linear and nonlinear stochastic autoregressive moving average (ARMA) model parameters, given a corrupt signal, using artificial recurrent neural networks. This new approach is a two-step approach in which the parameters of the deterministic...... part of the stochastic ARMA model are first estimated via a three-layer artificial neural network (deterministic estimation step) and then reestimated using the prediction error as one of the inputs to the artificial neural networks in an iterative algorithm (stochastic estimation step). The prediction...... of significant amounts of either dynamic or measurement noise in the output signal. The comparison between the deterministic and stochastic recurrent neural network approaches is furthered by applying both approaches to experimentally obtained renal blood pressure and flow signals....
Population density equations for stochastic processes with memory kernels.
Lai, Yi Ming; de Kamps, Marc
2017-06-01
We present a method for solving population density equations (PDEs)--a mean-field technique describing homogeneous populations of uncoupled neurons-where the populations can be subject to non-Markov noise for arbitrary distributions of jump sizes. The method combines recent developments in two different disciplines that traditionally have had limited interaction: computational neuroscience and the theory of random networks. The method uses a geometric binning scheme, based on the method of characteristics, to capture the deterministic neurodynamics of the population, separating the deterministic and stochastic process cleanly. We can independently vary the choice of the deterministic model and the model for the stochastic process, leading to a highly modular numerical solution strategy. We demonstrate this by replacing the master equation implicit in many formulations of the PDE formalism by a generalization called the generalized Montroll-Weiss equation-a recent result from random network theory-describing a random walker subject to transitions realized by a non-Markovian process. We demonstrate the method for leaky- and quadratic-integrate and fire neurons subject to spike trains with Poisson and gamma-distributed interspike intervals. We are able to model jump responses for both models accurately to both excitatory and inhibitory input under the assumption that all inputs are generated by one renewal process.
Infinite time interval backward stochastic differential equations with continuous coefficients.
Zong, Zhaojun; Hu, Feng
2016-01-01
In this paper, we study the existence theorem for [Formula: see text] [Formula: see text] solutions to a class of 1-dimensional infinite time interval backward stochastic differential equations (BSDEs) under the conditions that the coefficients are continuous and have linear growths. We also obtain the existence of a minimal solution. Furthermore, we study the existence and uniqueness theorem for [Formula: see text] [Formula: see text] solutions of infinite time interval BSDEs with non-uniformly Lipschitz coefficients. It should be pointed out that the assumptions of this result is weaker than that of Theorem 3.1 in Zong (Turkish J Math 37:704-718, 2013).
xSPDE: Extensible software for stochastic equations
Directory of Open Access Journals (Sweden)
Simon Kiesewetter
2016-01-01
Full Text Available We introduce an extensible software toolbox, xSPDE, for solving ordinary and partial stochastic differential equations. The toolbox makes extensive use of vector and parallel methods. Inputs are exceptionally simple, to reduce the learning curve, with default options for all of the many input parameters. The code calculates functional means, correlations and spectra, checks for errors in both time-step and sampling, and provides several choices of algorithm. Most aspects of the code, including the numerical algorithm, have a modular functional design to allow user modifications.
Variance estimates for transport in stochastic media by means of the master equation
International Nuclear Information System (INIS)
Pautz, S. D.; Franke, B. C.; Prinja, A. K.
2013-01-01
The master equation has been used to examine properties of transport in stochastic media. It has been shown previously that not only may the Levermore-Pomraning (LP) model be derived from the master equation for a description of ensemble-averaged transport quantities, but also that equations describing higher-order statistical moments may be obtained. We examine in greater detail the equations governing the second moments of the distribution of the angular fluxes, from which variances may be computed. We introduce a simple closure for these equations, as well as several models for estimating the variances of derived transport quantities. We revisit previous benchmarks for transport in stochastic media in order to examine the error of these new variance models. We find, not surprisingly, that the errors in these variance estimates are at least as large as the corresponding estimates of the average, and sometimes much larger. We also identify patterns in these variance estimates that may help guide the construction of more accurate models. (authors)
Stochastic modeling of mode interactions via linear parabolized stability equations
Ran, Wei; Zare, Armin; Hack, M. J. Philipp; Jovanovic, Mihailo
2017-11-01
Low-complexity approximations of the Navier-Stokes equations have been widely used in the analysis of wall-bounded shear flows. In particular, the parabolized stability equations (PSE) and Floquet theory have been employed to capture the evolution of primary and secondary instabilities in spatially-evolving flows. We augment linear PSE with Floquet analysis to formally treat modal interactions and the evolution of secondary instabilities in the transitional boundary layer via a linear progression. To this end, we leverage Floquet theory by incorporating the primary instability into the base flow and accounting for different harmonics in the flow state. A stochastic forcing is introduced into the resulting linear dynamics to model the effect of nonlinear interactions on the evolution of modes. We examine the H-type transition scenario to demonstrate how our approach can be used to model nonlinear effects and capture the growth of the fundamental and subharmonic modes observed in direct numerical simulations and experiments.
Nonparametric estimation of stochastic differential equations with sparse Gaussian processes.
García, Constantino A; Otero, Abraham; Félix, Paulo; Presedo, Jesús; Márquez, David G
2017-08-01
The application of stochastic differential equations (SDEs) to the analysis of temporal data has attracted increasing attention, due to their ability to describe complex dynamics with physically interpretable equations. In this paper, we introduce a nonparametric method for estimating the drift and diffusion terms of SDEs from a densely observed discrete time series. The use of Gaussian processes as priors permits working directly in a function-space view and thus the inference takes place directly in this space. To cope with the computational complexity that requires the use of Gaussian processes, a sparse Gaussian process approximation is provided. This approximation permits the efficient computation of predictions for the drift and diffusion terms by using a distribution over a small subset of pseudosamples. The proposed method has been validated using both simulated data and real data from economy and paleoclimatology. The application of the method to real data demonstrates its ability to capture the behavior of complex systems.
Energy Technology Data Exchange (ETDEWEB)
Heydari, M.H., E-mail: heydari@stu.yazd.ac.ir [Faculty of Mathematics, Yazd University, Yazd (Iran, Islamic Republic of); The Laboratory of Quantum Information Processing, Yazd University, Yazd (Iran, Islamic Republic of); Hooshmandasl, M.R., E-mail: hooshmandasl@yazd.ac.ir [Faculty of Mathematics, Yazd University, Yazd (Iran, Islamic Republic of); The Laboratory of Quantum Information Processing, Yazd University, Yazd (Iran, Islamic Republic of); Cattani, C., E-mail: ccattani@unisa.it [Department of Mathematics, University of Salerno, Via Ponte Don Melillo, 84084 Fisciano (Italy); Maalek Ghaini, F.M., E-mail: maalek@yazd.ac.ir [Faculty of Mathematics, Yazd University, Yazd (Iran, Islamic Republic of); The Laboratory of Quantum Information Processing, Yazd University, Yazd (Iran, Islamic Republic of)
2015-02-15
Because of the nonlinearity, closed-form solutions of many important stochastic functional equations are virtually impossible to obtain. Thus, numerical solutions are a viable alternative. In this paper, a new computational method based on the generalized hat basis functions together with their stochastic operational matrix of Itô-integration is proposed for solving nonlinear stochastic Itô integral equations in large intervals. In the proposed method, a new technique for computing nonlinear terms in such problems is presented. The main advantage of the proposed method is that it transforms problems under consideration into nonlinear systems of algebraic equations which can be simply solved. Error analysis of the proposed method is investigated and also the efficiency of this method is shown on some concrete examples. The obtained results reveal that the proposed method is very accurate and efficient. As two useful applications, the proposed method is applied to obtain approximate solutions of the stochastic population growth models and stochastic pendulum problem.
Population density equations for stochastic processes with memory kernels
Lai, Yi Ming; de Kamps, Marc
2017-06-01
We present a method for solving population density equations (PDEs)-a mean-field technique describing homogeneous populations of uncoupled neurons—where the populations can be subject to non-Markov noise for arbitrary distributions of jump sizes. The method combines recent developments in two different disciplines that traditionally have had limited interaction: computational neuroscience and the theory of random networks. The method uses a geometric binning scheme, based on the method of characteristics, to capture the deterministic neurodynamics of the population, separating the deterministic and stochastic process cleanly. We can independently vary the choice of the deterministic model and the model for the stochastic process, leading to a highly modular numerical solution strategy. We demonstrate this by replacing the master equation implicit in many formulations of the PDE formalism by a generalization called the generalized Montroll-Weiss equation—a recent result from random network theory—describing a random walker subject to transitions realized by a non-Markovian process. We demonstrate the method for leaky- and quadratic-integrate and fire neurons subject to spike trains with Poisson and gamma-distributed interspike intervals. We are able to model jump responses for both models accurately to both excitatory and inhibitory input under the assumption that all inputs are generated by one renewal process.
Polynomial asymptotic stability of damped stochastic differential equations
Directory of Open Access Journals (Sweden)
John Appleby
2004-08-01
Full Text Available The paper studies the polynomial convergence of solutions of a scalar nonlinear It\\^{o} stochastic differential equation\\[dX(t = -f(X(t\\,dt + \\sigma(t\\,dB(t\\] where it is known, {\\it a priori}, that $\\lim_{t\\rightarrow\\infty} X(t=0$, a.s. The intensity of the stochastic perturbation $\\sigma$ is a deterministic, continuous and square integrable function, which tends to zero more quickly than a polynomially decaying function. The function $f$ obeys $\\lim_{x\\rightarrow 0}\\mbox{sgn}(xf(x/|x|^\\beta = a$, for some $\\beta>1$, and $a>0$.We study two asymptotic regimes: when $\\sigma$ tends to zero sufficiently quickly the polynomial decay rate of solutions is the same as for the deterministic equation (when $\\sigma\\equiv0$. When $\\sigma$ decays more slowly, a weaker almost sure polynomial upper bound on the decay rate of solutions is established. Results which establish the necessity for $\\sigma$ to decay polynomially in order to guarantee the almost sure polynomial decay of solutions are also proven.
Hsieh, Chang-Yu; Cao, Jianshu
2018-01-01
We extend a standard stochastic theory to study open quantum systems coupled to a generic quantum environment. We exemplify the general framework by studying a two-level quantum system coupled bilinearly to the three fundamental classes of non-interacting particles: bosons, fermions, and spins. In this unified stochastic approach, the generalized stochastic Liouville equation (SLE) formally captures the exact quantum dissipations when noise variables with appropriate statistics for different bath models are applied. Anharmonic effects of a non-Gaussian bath are precisely encoded in the bath multi-time correlation functions that noise variables have to satisfy. Starting from the SLE, we devise a family of generalized hierarchical equations by averaging out the noise variables and expand bath multi-time correlation functions in a complete basis of orthonormal functions. The general hierarchical equations constitute systems of linear equations that provide numerically exact simulations of quantum dynamics. For bosonic bath models, our general hierarchical equation of motion reduces exactly to an extended version of hierarchical equation of motion which allows efficient simulation for arbitrary spectral densities and temperature regimes. Similar efficiency and flexibility can be achieved for the fermionic bath models within our formalism. The spin bath models can be simulated with two complementary approaches in the present formalism. (I) They can be viewed as an example of non-Gaussian bath models and be directly handled with the general hierarchical equation approach given their multi-time correlation functions. (II) Alternatively, each bath spin can be first mapped onto a pair of fermions and be treated as fermionic environments within the present formalism.
Müller, Eike H; Scheichl, Rob; Shardlow, Tony
2015-04-08
This paper applies several well-known tricks from the numerical treatment of deterministic differential equations to improve the efficiency of the multilevel Monte Carlo (MLMC) method for stochastic differential equations (SDEs) and especially the Langevin equation. We use modified equations analysis as an alternative to strong-approximation theory for the integrator, and we apply this to introduce MLMC for Langevin-type equations with integrators based on operator splitting. We combine this with extrapolation and investigate the use of discrete random variables in place of the Gaussian increments, which is a well-known technique for the weak approximation of SDEs. We show that, for small-noise problems, discrete random variables can lead to an increase in efficiency of almost two orders of magnitude for practical levels of accuracy.
Fractional Stochastic Differential Equations Satisfying Fluctuation-Dissipation Theorem
Li, Lei; Liu, Jian-Guo; Lu, Jianfeng
2017-10-01
We propose in this work a fractional stochastic differential equation (FSDE) model consistent with the over-damped limit of the generalized Langevin equation model. As a result of the `fluctuation-dissipation theorem', the differential equations driven by fractional Brownian noise to model memory effects should be paired with Caputo derivatives, and this FSDE model should be understood in an integral form. We establish the existence of strong solutions for such equations and discuss the ergodicity and convergence to Gibbs measure. In the linear forcing regime, we show rigorously the algebraic convergence to Gibbs measure when the `fluctuation-dissipation theorem' is satisfied, and this verifies that satisfying `fluctuation-dissipation theorem' indeed leads to the correct physical behavior. We further discuss possible approaches to analyze the ergodicity and convergence to Gibbs measure in the nonlinear forcing regime, while leave the rigorous analysis for future works. The FSDE model proposed is suitable for systems in contact with heat bath with power-law kernel and subdiffusion behaviors.
Inviscid limit of stochastic damped 2D Navier–Stokes equations
International Nuclear Information System (INIS)
Bessaih, Hakima; Ferrario, Benedetta
2014-01-01
We consider the inviscid limit of the stochastic damped 2D Navier–Stokes equations. We prove that, when the viscosity vanishes, the stationary solution of the stochastic damped Navier–Stokes equations converges to a stationary solution of the stochastic damped Euler equation and that the rate of dissipation of enstrophy converges to zero. In particular, this limit obeys an enstrophy balance. The rates are computed with respect to a limit measure of the unique invariant measure of the stochastic damped Navier–Stokes equations. (paper)
A Newton method for the resolution of steady stochastic Navier-Stokes equations
International Nuclear Information System (INIS)
Le Maitre, O.
2009-01-01
We present a Newton method to compute the stochastic solution of the steady incompressible Navier-Stokes equations with random data (boundary conditions, forcing term, fluid properties). The method assumes a spectral discretization at the stochastic level involving a orthogonal basis of random functionals (such as Polynomial Chaos or stochastic multi-wavelets bases). The Newton method uses the unsteady equations to derive a linear equation for the stochastic Newton increments. This linear equation is subsequently solved following a matrix-free strategy, where the iterations consist in performing integrations of the linearized unsteady Navier-Stokes equations, with an appropriate time scheme to allow for a decoupled integration of the stochastic modes. Various examples are provided to demonstrate the efficiency of the method in determining stochastic steady solution, even for regimes where it is likely unstable. (authors)
Developing Itô stochastic differential equation models for neuronal signal transduction pathways.
Manninen, Tiina; Linne, Marja-Leena; Ruohonen, Keijo
2006-08-01
Mathematical modeling and simulation of dynamic biochemical systems are receiving considerable attention due to the increasing availability of experimental knowledge of complex intracellular functions. In addition to deterministic approaches, several stochastic approaches have been developed for simulating the time-series behavior of biochemical systems. The problem with stochastic approaches, however, is the larger computational time compared to deterministic approaches. It is therefore necessary to study alternative ways to incorporate stochasticity and to seek approaches that reduce the computational time needed for simulations, yet preserve the characteristic behavior of the system in question. In this work, we develop a computational framework based on the Itô stochastic differential equations for neuronal signal transduction networks. There are several different ways to incorporate stochasticity into deterministic differential equation models and to obtain Itô stochastic differential equations. Two of the developed models are found most suitable for stochastic modeling of neuronal signal transduction. The best models give stable responses which means that the variances of the responses with time are not increasing and negative concentrations are avoided. We also make a comparative analysis of different kinds of stochastic approaches, that is the Itô stochastic differential equations, the chemical Langevin equation, and the Gillespie stochastic simulation algorithm. Different kinds of stochastic approaches can be used to produce similar responses for the neuronal protein kinase C signal transduction pathway. The fine details of the responses vary slightly, depending on the approach and the parameter values. However, when simulating great numbers of chemical species, the Gillespie algorithm is computationally several orders of magnitude slower than the Itô stochastic differential equations and the chemical Langevin equation. Furthermore, the chemical
Monte-Carlo simulation of a stochastic differential equation
Arif, ULLAH; Majid, KHAN; M, KAMRAN; R, KHAN; Zhengmao, SHENG
2017-12-01
For solving higher dimensional diffusion equations with an inhomogeneous diffusion coefficient, Monte Carlo (MC) techniques are considered to be more effective than other algorithms, such as finite element method or finite difference method. The inhomogeneity of diffusion coefficient strongly limits the use of different numerical techniques. For better convergence, methods with higher orders have been kept forward to allow MC codes with large step size. The main focus of this work is to look for operators that can produce converging results for large step sizes. As a first step, our comparative analysis has been applied to a general stochastic problem. Subsequently, our formulization is applied to the problem of pitch angle scattering resulting from Coulomb collisions of charge particles in the toroidal devices.
Real and Complex Turbulence for the Stochastic Burgers Equation
Neate, A
2004-01-01
The inviscid limit of Burgers equation, with body forces white noise in time, is discussed in terms of the level surfaces of the minimising Hamilton-Jacobi function and the classical mechanical caustic. Presurfaces and precaustics are introduced by using the classical mechanical flow map. When the prelevel surface touches the precaustic, the geometry (number of cusps) on the level surface changes infinitely rapidly causing `real turbulence' (Davies, Truman and Zhao). Using an idea of Felix Klein, it is shown that the geometry (number of swallowtails) on the caustic also changes infinitely rapidly when the real part of the precaustic touches its complex counterpart, which we call `complex turbulence'. These two new kinds of turbulence are both inherently stochastic in nature. A complete analysis of this problem is given in terms of a reduced (one dimensional) action function. This characterises which parts of the original caustic are singular - an old problem in applied mathematics relevant for our `elementary...
Numerical approximation of random periodic solutions of stochastic differential equations
Feng, Chunrong; Liu, Yu; Zhao, Huaizhong
2017-10-01
In this paper, we discuss the numerical approximation of random periodic solutions of stochastic differential equations (SDEs) with multiplicative noise. We prove the existence of the random periodic solution as the limit of the pull-back flow when the starting time tends to -∞ along the multiple integrals of the period. As the random periodic solution is not explicitly constructible, it is useful to study the numerical approximation. We discretise the SDE using the Euler-Maruyama scheme and modified Milstein scheme. Subsequently, we obtain the existence of the random periodic solution as the limit of the pull-back of the discretised SDE. We prove that the latter is an approximated random periodic solution with an error to the exact one at the rate of √{Δ t} in the mean square sense in Euler-Maruyama method and Δ t in the Milstein method. We also obtain the weak convergence result for the approximation of the periodic measure.
Modeling the euglycemic hyperinsulinemic clamp by stochastic differential equations.
Picchini, Umberto; Ditlevsen, Susanne; De Gaetano, Andrea
2006-11-01
The Euglycemic Hyperinsulinemic Clamp (EHC) is the most widely used experimental procedure for the determination of insulin sensitivity. In the present study, 16 subjects with BMI between 18.5 and 63.6 kg/m(2) have been studied with a long-duration (5 hours) EHC. In order to explain the oscillations of glycemia occurring in response to the hyperinsulinization and to the continuous glucose infusion at varying speeds, we first hypothesized a system of ordinary differential equations (ODEs), with limited success. We then extended the model and represented the experiment using a system of stochastic differential equations (SDEs). The latter allow for distinction between (i) random variation imputable to observation error and (ii) system noise (intrinsic variability of the metabolic system), due to a variety of influences which change over time. The stochastic model of the EHC was fitted to data and the system noise was estimated by means of a (simulated) maximum likelihood procedure, for a series of different hypothetical measurement error values. We showed that, for the whole range of reasonable measurement error values: (i) the system noise estimates are non-negligible; and (ii) these estimates are robust to changes in the likely value of the measurement error. Explicit expression of system noise is physiologically relevant in this case, since glucose uptake rate is known to be affected by a host of additive influences, usually neglected when modeling metabolism. While in some of the studied subjects system noise appeared to only marginally affect the dynamics, in others the system appeared to be driven more by the erratic oscillations in tissue glucose transport rather than by the overall glucose-insulin control system. It is possible that the quantitative relevance of the unexpressed effects (system noise) should be considered in other physiological situations, represented so far only with deterministic models.
Some additional remarks on the cumulant expansion for linear stochastic differential equations
Roerdink, J.B.T.M.
1984-01-01
We summarize our previous results on cumular expasions for linear stochastic differential equations with correlated multipliclative and additive noise. The application of the general formulas to equations with statistically independent multiplicative and additive noise is reconsidered in detail,
Some Additional Remarks on the Cumulant Expansion for Linear Stochastic Differential Equations
Roerdink, J.B.T.M.
1984-01-01
We summarize our previous results on cumulant expansions for linear stochastic differential equations with correlated multipliclative and additive noise. The application of the general formulas to equations with statistically independent multiplicative and additive noise is reconsidered in detail,
Frank, T D; Beek, P J
2001-08-01
Recently, Küchler and Mensch [Stochastics Stochastics Rep. 40, 23 (1992)] derived exact stationary probability densities for linear stochastic delay differential equations. This paper presents an alternative derivation of these solutions by means of the Fokker-Planck approach introduced by Guillouzic [Phys. Rev. E 59, 3970 (1999); 61, 4906 (2000)]. Applications of this approach, which is argued to have greater generality, are discussed in the context of stochastic models for population growth and tracking movements.
Gaudreault, Mathieu; Drolet, François; Viñals, Jorge
2010-11-01
Analytical expressions for pitchfork and Hopf bifurcation thresholds are given for a nonlinear stochastic differential delay equation with feedback. Our results assume that the delay time τ is small compared to other characteristic time scales, not a significant limitation close to the bifurcation line. A pitchfork bifurcation line is found, the location of which depends on the conditional average , where x(t) is the dynamical variable. This conditional probability incorporates the combined effect of fluctuation correlations and delayed feedback. We also find a Hopf bifurcation line which is obtained by a multiple scale expansion around the oscillatory solution near threshold. We solve the Fokker-Planck equation associated with the slowly varying amplitudes and use it to determine the threshold location. In both cases, the predicted bifurcation lines are in excellent agreement with a direct numerical integration of the governing equations. Contrary to the known case involving no delayed feedback, we show that the stochastic bifurcation lines are shifted relative to the deterministic limit and hence that the interaction between fluctuation correlations and delay affect the stability of the solutions of the model equation studied.
Exponential p-stability of impulsive stochastic differential equations with delays
International Nuclear Information System (INIS)
Yang Zhiguo; Xu Daoyi; Xiang Li
2006-01-01
In this Letter, we establish a method to study the exponential p-stability of the zero solution of impulsive stochastic differential equations with delays. By establishing an L-operator inequality and using the properties of M-cone and stochastic analysis technique, we obtain some new conditions ensuring the exponential p-stability of the zero solution of impulsive stochastic differential equations with delays. Two illustrative examples have been provided to show the effectiveness of our results
Chadha, Alka; Bora, Swaroop Nandan
2017-11-01
This paper studies the existence, uniqueness, and exponential stability in mean square for the mild solution of neutral second order stochastic partial differential equations with infinite delay and Poisson jumps. By utilizing the Banach fixed point theorem, first the existence and uniqueness of the mild solution of neutral second order stochastic differential equations is established. Then, the mean square exponential stability for the mild solution of the stochastic system with Poisson jumps is obtained with the help of an established integral inequality.
Riemann integral of a random function and the parabolic equation with a general stochastic measure
Radchenko, Vadym
2012-01-01
For stochastic parabolic equation driven by a general stochastic measure, the weak solution is obtained. The integral of a random function in the equation is considered as a limit in probability of Riemann integral sums. Basic properties of such integrals are studied in the paper.
On solutions of neutral stochastic delay Volterra equations with singular kernels
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Xiaotai Wu
2012-08-01
Full Text Available In this paper, existence, uniqueness and continuity of the adapted solutions for neutral stochastic delay Volterra equations with singular kernels are discussed. In addition, continuous dependence on the initial date is also investigated. Finally, stochastic Volterra equation with the kernel of fractional Brownian motion is studied to illustrate the effectiveness of our results.
Hyperbolic white noise functional solutions of Wick-type stochastic compound KdV-Burgers equations
International Nuclear Information System (INIS)
Han Xiu; Xie Yingchao
2009-01-01
Variable coefficient and Wick-type stochastic compound KdV-Burgers equations are investigated. By using white noise analysis, Hermite transform and the hyperbolic function method, we obtain a number of Wick versions of hyperbolic white noise functional solutions and hyperbolic function solutions for Wick-type stochastic and variable coefficient compound KdV-Burgers equations, respectively.
Semilinear stochastic equations in a Hilbert space with a fractional Brownian motion
Czech Academy of Sciences Publication Activity Database
Duncan, T. E.; Maslowski, Bohdan; Pasik-Duncan, B.
2009-01-01
Roč. 40, č. 6 (2009), s. 2286-2315 ISSN 0036-1410 R&D Projects: GA ČR GA201/07/0237 Institutional research plan: CEZ:AV0Z10190503 Keywords : semilinear stochastic equations * fractional Brownian motion * stochastic partial differential equations * absolute continuity of measures Subject RIV: BA - General Mathematics Impact factor: 1.649, year: 2009
Horowitz, Jordan M
2015-07-28
The stochastic thermodynamics of a dilute, well-stirred mixture of chemically reacting species is built on the stochastic trajectories of reaction events obtained from the chemical master equation. However, when the molecular populations are large, the discrete chemical master equation can be approximated with a continuous diffusion process, like the chemical Langevin equation or low noise approximation. In this paper, we investigate to what extent these diffusion approximations inherit the stochastic thermodynamics of the chemical master equation. We find that a stochastic-thermodynamic description is only valid at a detailed-balanced, equilibrium steady state. Away from equilibrium, where there is no consistent stochastic thermodynamics, we show that one can still use the diffusive solutions to approximate the underlying thermodynamics of the chemical master equation.
Stochastic solutions to the Schrodinger equation for fermions
International Nuclear Information System (INIS)
Arnow, D.M.
1981-01-01
An exact stochastic method has been developed for generating the antisymmetric eigensolution of lowest index and its associated eigenvalue for the Schrodinger wave equation in 3N dimensions. The method is called the Green's function Monte Carlo method for fermions (FGFMC) because it is based on a Monte Carlo solution to the integral form of the Schrodinger equation (using Green's function) and because it is the fermion class of particles in physics which require antisymmetric solutions. The solution consists of two sets of 3N-dimensional points, [R/sub j/ + ] and [R/sub j/ - ], distributed by density functions psi + and psi - , whose difference, psi + -psi - , is proportional to the eigensolution, psi/sub F/. The FGFMC method is successfully applied to a one dimensional problem and a nine dimensional problem, the results of which are presented here. These results demonstrate that this method can be successfully applied to small physical problems on medium-scale computing machines. The key to this success was the transformation of the problem from exponential to linear cost as a function of accuracy. The strong dependence on dimensionality, however, currently results in an exponential cost as a function of problem size, and this, until overcome, imposes a severe barrier to calculations on large systems
Climate Change and Integrodifference Equations in a Stochastic Environment.
Bouhours, Juliette; Lewis, Mark A
2016-09-01
Climate change impacts population distributions, forcing some species to migrate poleward if they are to survive and keep up with the suitable habitat that is shifting with the temperature isoclines. Previous studies have analysed whether populations have the capacity to keep up with shifting temperature isoclines, and have mathematically determined the combination of growth and dispersal that is needed to achieve this. However, the rate of isocline movement can be highly variable, with much uncertainty associated with yearly shifts. The same is true for population growth rates. Growth rates can be variable and uncertain, even within suitable habitats for growth. In this paper, we reanalyse the question of population persistence in the context of the uncertainty and variability in isocline shifts and rates of growth. Specifically, we employ a stochastic integrodifference equation model on a patch of suitable habitat that shifts poleward at a random rate. We derive a metric describing the asymptotic growth rate of the linearised operator of the stochastic model. This metric yields a threshold criterion for population persistence. We demonstrate that the variability in the yearly shift and in the growth rate has a significant negative effect on the persistence in the sense that it decreases the threshold criterion for population persistence. Mathematically, we show how the persistence metric can be connected to the principal eigenvalue problem for a related integral operator, at least for the case where isocline shifting speed is deterministic. Analysis of dynamics for the case where the dispersal kernel is Gaussian leads to the existence of a critical shifting speed, above which the population will go extinct, and below which the population will persist. This leads to clear bounds on rate of environmental change if the population is to persist. Finally, we illustrate our different results for butterfly population using numerical simulations and demonstrate how
DEFF Research Database (Denmark)
Chon, K H; Hoyer, D; Armoundas, A A
1999-01-01
In this study, we introduce a new approach for estimating linear and nonlinear stochastic autoregressive moving average (ARMA) model parameters, given a corrupt signal, using artificial recurrent neural networks. This new approach is a two-step approach in which the parameters of the deterministic...... error is obtained by subtracting the corrupt signal of the estimated ARMA model obtained via the deterministic estimation step from the system output response. We present computer simulation examples to show the efficacy of the proposed stochastic recurrent neural network approach in obtaining accurate...... model predictions. Furthermore, we compare the performance of the new approach to that of the deterministic recurrent neural network approach. Using this simple two-step procedure, we obtain more robust model predictions than with the deterministic recurrent neural network approach despite the presence...
International Nuclear Information System (INIS)
Gambetta, Jay; Wiseman, H.M.
2002-01-01
Do stochastic Schroedinger equations, also known as unravelings, have a physical interpretation? In the Markovian limit, where the system on average obeys a master equation, the answer is yes. Markovian stochastic Schroedinger equations generate quantum trajectories for the system state conditioned on continuously monitoring the bath. For a given master equation, there are many different unravelings, corresponding to different sorts of measurement on the bath. In this paper we address the non-Markovian case, and in particular the sort of stochastic Schroedinger equation introduced by Strunz, Diosi, and Gisin [Phys. Rev. Lett. 82, 1801 (1999)]. Using a quantum-measurement theory approach, we rederive their unraveling that involves complex-valued Gaussian noise. We also derive an unraveling involving real-valued Gaussian noise. We show that in the Markovian limit, these two unravelings correspond to heterodyne and homodyne detection, respectively. Although we use quantum-measurement theory to define these unravelings, we conclude that the stochastic evolution of the system state is not a true quantum trajectory, as the identity of the state through time is a fiction
Granita, Bahar, A.
2015-03-01
This paper discusses on linear birth and death with immigration and emigration (BIDE) process to stochastic differential equation (SDE) model. Forward Kolmogorov equation in continuous time Markov chain (CTMC) with a central-difference approximation was used to find Fokker-Planckequation corresponding to a diffusion process having the stochastic differential equation of BIDE process. The exact solution, mean and variance function of BIDE process was found.
International Nuclear Information System (INIS)
Granita; Bahar, A.
2015-01-01
This paper discusses on linear birth and death with immigration and emigration (BIDE) process to stochastic differential equation (SDE) model. Forward Kolmogorov equation in continuous time Markov chain (CTMC) with a central-difference approximation was used to find Fokker-Planckequation corresponding to a diffusion process having the stochastic differential equation of BIDE process. The exact solution, mean and variance function of BIDE process was found
Energy Technology Data Exchange (ETDEWEB)
Granita, E-mail: granitafc@gmail.com [Dept. Mathematical Education, State Islamic University of Sultan Syarif Kasim Riau, 28293 Indonesia and Dept. of Mathematical Science, Universiti Teknologi Malaysia, 81310,Johor (Malaysia); Bahar, A. [Dept. of Mathematical Science, Universiti Teknologi Malaysia, 81310,Johor Malaysia and UTM Center for Industrial and Applied Mathematics (UTM-CIAM) (Malaysia)
2015-03-09
This paper discusses on linear birth and death with immigration and emigration (BIDE) process to stochastic differential equation (SDE) model. Forward Kolmogorov equation in continuous time Markov chain (CTMC) with a central-difference approximation was used to find Fokker-Planckequation corresponding to a diffusion process having the stochastic differential equation of BIDE process. The exact solution, mean and variance function of BIDE process was found.
A New Control Paradigm for Stochastic Differential Equations
Schmid, Matthias J. A.
This study presents a novel comprehensive approach to the control of dynamic systems under uncertainty governed by stochastic differential equations (SDEs). Large Deviations (LD) techniques are employed to arrive at a control law for a large class of nonlinear systems minimizing sample path deviations. Thereby, a paradigm shift is suggested from point-in-time to sample path statistics on function spaces. A suitable formal control framework which leverages embedded Freidlin-Wentzell theory is proposed and described in detail. This includes the precise definition of the control objective and comprises an accurate discussion of the adaptation of the Freidlin-Wentzell theorem to the particular situation. The new control design is enabled by the transformation of an ill-posed control objective into a well-conditioned sequential optimization problem. A direct numerical solution process is presented using quadratic programming, but the emphasis is on the development of a closed-form expression reflecting the asymptotic deviation probability of a particular nominal path. This is identified as the key factor in the success of the new paradigm. An approach employing the second variation and the differential curvature of the effective action is suggested for small deviation channels leading to the Jacobi field of the rate function and the subsequently introduced Jacobi field performance measure. This closed-form solution is utilized in combination with the supplied parametrization of the objective space. For the first time, this allows for an LD based control design applicable to a large class of nonlinear systems. Thus, Minimum Large Deviations (MLD) control is effectively established in a comprehensive structured framework. The construction of the new paradigm is completed by an optimality proof for the Jacobi field performance measure, an interpretive discussion, and a suggestion for efficient implementation. The potential of the new approach is exhibited by its extension
A stochastic differential equation model of diurnal cortisol patterns
Brown, E. N.; Meehan, P. M.; Dempster, A. P.
2001-01-01
Circadian modulation of episodic bursts is recognized as the normal physiological pattern of diurnal variation in plasma cortisol levels. The primary physiological factors underlying these diurnal patterns are the ultradian timing of secretory events, circadian modulation of the amplitude of secretory events, infusion of the hormone from the adrenal gland into the plasma, and clearance of the hormone from the plasma by the liver. Each measured plasma cortisol level has an error arising from the cortisol immunoassay. We demonstrate that all of these three physiological principles can be succinctly summarized in a single stochastic differential equation plus measurement error model and show that physiologically consistent ranges of the model parameters can be determined from published reports. We summarize the model parameters in terms of the multivariate Gaussian probability density and establish the plausibility of the model with a series of simulation studies. Our framework makes possible a sensitivity analysis in which all model parameters are allowed to vary simultaneously. The model offers an approach for simultaneously representing cortisol's ultradian, circadian, and kinetic properties. Our modeling paradigm provides a framework for simulation studies and data analysis that should be readily adaptable to the analysis of other endocrine hormone systems.
Stochastic Growth Theory of Spatially-Averaged Distributions of Langmuir Fields in Earth's Foreshock
Boshuizen, Christopher R.; Cairns, Iver H.; Robinson, P. A.
2001-01-01
Langmuir-like waves in the foreshock of Earth are characteristically bursty and irregular, and are the subject of a number of recent studies. Averaged over the foreshock, it is observed that the probability distribution is power-law P(bar)(log E) in the wave field E with the bar denoting this averaging over position, In this paper it is shown that stochastic growth theory (SGT) can explain a power-law spatially-averaged distributions P(bar)(log E), when the observed power-law variations of the mean and standard deviation of log E with position are combined with the log normal statistics predicted by SGT at each location.
Solving Langevin equation with the stochastic algebraically correlated noise
International Nuclear Information System (INIS)
Ploszajczak, M.; Srokowski, T.
1996-01-01
Long time tail in the velocity and force autocorrelation function has been found recently in the molecular dynamics simulations of the peripheral collisions of ions. Simulation of those slowly decaying correlations in the stochastic transport theory requires the development of new methods of generating stochastic force of arbitrarily long correlation times. The Markovian process and the multidimensional Kangaroo process which permit describing various algebraic correlated stochastic processes are proposed. (author)
Random Attractors for Stochastic Retarded Reaction-Diffusion Equations on Unbounded Domains
Directory of Open Access Journals (Sweden)
Xiaoquan Ding
2013-01-01
Full Text Available This paper is devoted to a stochastic retarded reaction-diffusion equation on all d-dimensional space with additive white noise. We first show that the stochastic retarded reaction-diffusion equation generates a random dynamical system by transforming this stochastic equation into a random one through a tempered stationary random homeomorphism. Then, we establish the existence of a random attractor for the random equation. And the existence of a random attractor for the stochastic equation follows from the conjugation relation between two random dynamical systems. The pullback asymptotic compactness is proved by uniform estimates on solutions for large space and time variables. These estimates are obtained by a cut-off technique.
Approximate Dual Averaging Method for Multiagent Saddle-Point Problems with Stochastic Subgradients
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Deming Yuan
2014-01-01
Full Text Available This paper considers the problem of solving the saddle-point problem over a network, which consists of multiple interacting agents. The global objective function of the problem is a combination of local convex-concave functions, each of which is only available to one agent. Our main focus is on the case where the projection steps are calculated approximately and the subgradients are corrupted by some stochastic noises. We propose an approximate version of the standard dual averaging method and show that the standard convergence rate is preserved, provided that the projection errors decrease at some appropriate rate and the noises are zero-mean and have bounded variance.
Almost Surely Exponential Stability of Numerical Solutions for Stochastic Pantograph Equations
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Shaobo Zhou
2014-01-01
Full Text Available Our effort is to develop a criterion on almost surely exponential stability of numerical solution to stochastic pantograph differential equations, with the help of the discrete semimartingale convergence theorem and the technique used in stable analysis of the exact solution. We will prove that the Euler-Maruyama (EM method can preserve almost surely exponential stability of stochastic pantograph differential equations under the linear growth conditions. And the backward EM method can reproduce almost surely exponential stability for highly nonlinear stochastic pantograph differential equations. A highly nonlinear example is provided to illustrate the main theory.
Wang, Xiaohu; Lu, Kening; Wang, Bixiang
2018-01-01
In this paper, we study the Wong-Zakai approximations given by a stationary process via the Wiener shift and their associated long term behavior of the stochastic reaction-diffusion equation driven by a white noise. We first prove the existence and uniqueness of tempered pullback attractors for the Wong-Zakai approximations of stochastic reaction-diffusion equation. Then, we show that the attractors of Wong-Zakai approximations converges to the attractor of the stochastic reaction-diffusion equation for both additive and multiplicative noise.
Wong-Zakai approximations and center manifolds of stochastic differential equations
Shen, Jun; Lu, Kening
2017-10-01
In this paper, we study the Wong-Zakai approximations given by a stationary process via the Wiener shift and their associated dynamics of the stochastic differential equation driven by a l-dimensional Brownian motion. We prove that the solutions of Wong-Zakai approximations converge in the mean square to the solutions of the Stratonovich stochastic differential equation. We also show that for a simple multiplicative noise, the center-manifold of the Wong-Zakai approximations converges to the center-manifold of the Stratonovich stochastic differential equation.
Klim, Søren; Mortensen, Stig Bousgaard; Kristensen, Niels Rode; Overgaard, Rune Viig; Madsen, Henrik
2009-06-01
The extension from ordinary to stochastic differential equations (SDEs) in pharmacokinetic and pharmacodynamic (PK/PD) modelling is an emerging field and has been motivated in a number of articles [N.R. Kristensen, H. Madsen, S.H. Ingwersen, Using stochastic differential equations for PK/PD model development, J. Pharmacokinet. Pharmacodyn. 32 (February(1)) (2005) 109-141; C.W. Tornøe, R.V. Overgaard, H. Agersø, H.A. Nielsen, H. Madsen, E.N. Jonsson, Stochastic differential equations in NONMEM: implementation, application, and comparison with ordinary differential equations, Pharm. Res. 22 (August(8)) (2005) 1247-1258; R.V. Overgaard, N. Jonsson, C.W. Tornøe, H. Madsen, Non-linear mixed-effects models with stochastic differential equations: implementation of an estimation algorithm, J. Pharmacokinet. Pharmacodyn. 32 (February(1)) (2005) 85-107; U. Picchini, S. Ditlevsen, A. De Gaetano, Maximum likelihood estimation of a time-inhomogeneous stochastic differential model of glucose dynamics, Math. Med. Biol. 25 (June(2)) (2008) 141-155]. PK/PD models are traditionally based ordinary differential equations (ODEs) with an observation link that incorporates noise. This state-space formulation only allows for observation noise and not for system noise. Extending to SDEs allows for a Wiener noise component in the system equations. This additional noise component enables handling of autocorrelated residuals originating from natural variation or systematic model error. Autocorrelated residuals are often partly ignored in PK/PD modelling although violating the hypothesis for many standard statistical tests. This article presents a package for the statistical program R that is able to handle SDEs in a mixed-effects setting. The estimation method implemented is the FOCE(1) approximation to the population likelihood which is generated from the individual likelihoods that are approximated using the Extended Kalman Filter's one-step predictions.
Stochastic optimal control in infinite dimension dynamic programming and HJB equations
Fabbri, Giorgio; Święch, Andrzej
2017-01-01
Providing an introduction to stochastic optimal control in infinite dimension, this book gives a complete account of the theory of second-order HJB equations in infinite-dimensional Hilbert spaces, focusing on its applicability to associated stochastic optimal control problems. It features a general introduction to optimal stochastic control, including basic results (e.g. the dynamic programming principle) with proofs, and provides examples of applications. A complete and up-to-date exposition of the existing theory of viscosity solutions and regular solutions of second-order HJB equations in Hilbert spaces is given, together with an extensive survey of other methods, with a full bibliography. In particular, Chapter 6, written by M. Fuhrman and G. Tessitore, surveys the theory of regular solutions of HJB equations arising in infinite-dimensional stochastic control, via BSDEs. The book is of interest to both pure and applied researchers working in the control theory of stochastic PDEs, and in PDEs in infinite ...
An energy-conserving method for stochastic Maxwell equations with multiplicative noise
Hong, Jialin; Ji, Lihai; Zhang, Liying; Cai, Jiaxiang
2017-12-01
In this paper, it is shown that three-dimensional stochastic Maxwell equations with multiplicative noise are stochastic Hamiltonian partial differential equations possessing a geometric structure (i.e. stochastic multi-symplectic conservation law), and the energy of system is a conservative quantity almost surely. We propose a stochastic multi-symplectic energy-conserving method for the equations by using the wavelet collocation method in space and stochastic symplectic method in time. Numerical experiments are performed to verify the excellent abilities of the proposed method in providing accurate solution and preserving energy. The mean square convergence result of the method in temporal direction is tested numerically, and numerical comparisons with finite difference method are also investigated.
Kleinert, H; Zatloukal, V
2013-11-01
The statistics of rare events, the so-called black-swan events, is governed by non-Gaussian distributions with heavy power-like tails. We calculate the Green functions of the associated Fokker-Planck equations and solve the related stochastic differential equations. We also discuss the subject in the framework of path integration.
Zollanvari, Amin; Dougherty, Edward R
2016-12-01
In classification, prior knowledge is incorporated in a Bayesian framework by assuming that the feature-label distribution belongs to an uncertainty class of feature-label distributions governed by a prior distribution. A posterior distribution is then derived from the prior and the sample data. An optimal Bayesian classifier (OBC) minimizes the expected misclassification error relative to the posterior distribution. From an application perspective, prior construction is critical. The prior distribution is formed by mapping a set of mathematical relations among the features and labels, the prior knowledge, into a distribution governing the probability mass across the uncertainty class. In this paper, we consider prior knowledge in the form of stochastic differential equations (SDEs). We consider a vector SDE in integral form involving a drift vector and dispersion matrix. Having constructed the prior, we develop the optimal Bayesian classifier between two models and examine, via synthetic experiments, the effects of uncertainty in the drift vector and dispersion matrix. We apply the theory to a set of SDEs for the purpose of differentiating the evolutionary history between two species.
Directory of Open Access Journals (Sweden)
2006-01-01
Full Text Available The nonlinear alternative of Leray-Schauder type is used to investigate the existence of solutions for first-order semilinear stochastic functional differential equations in Hilbert spaces.
Non-Intrusive Solution of Stochastic and Parametric Equations
Matthies, Hermann
2015-01-07
Many problems depend on parameters, which may be a finite set of numerical values, or mathematically more complicated objects like for example processes or fields. We address the situation where we have an equation which depends on parameters; stochastic equations are a special case of such parametric problems where the parameters are elements from a probability space. One common way to represent this dependability on parameters is by evaluating the state (or solution) of the system under investigation for different values of the parameters. But often one wants to evaluate the solution quickly for a new set of parameters where it has not been sampled. In this situation it may be advantageous to express the parameter dependent solution with an approximation which allows for rapid evaluation of the solution. Such approximations are also called proxy or surrogate models, response functions, or emulators. All these methods may be seen as functional approximations—representations of the solution by an “easily computable” function of the parameters, as opposed to pure samples. The most obvious methods of approximation used are based on interpolation, in this context often labelled as collocation. In the frequent situation where one has a “solver” for the equation for a given parameter value, i.e. a software component or a program, it is evident that this can be used to independently—if desired in parallel—solve for all the parameter values which subsequently may be used either for the interpolation or in the quadrature for the projection. Such methods are therefore uncoupled for each parameter value, and they additionally often carry the label “non-intrusive”. Without much argument all other methods— which produce a coupled system of equations–are almost always labelled as “intrusive”, meaning that one cannot use the original solver. We want to show here that this not necessarily the case. Another approach is to choose some other projection onto
Agarwal, Ankush; De Marco, Stefano; GOBET, Emmanuel; López-Salas, José; Noubiagain, Fanny; Zhou, Alexandre
2018-01-01
We introduce a new class of anticipative backward stochastic differential equations with a dependence of McKean type on the law of the solution, that we name MKABSDE. We provide existence and uniqueness results in a general framework with relaxed regularity assumptions on the parameters. We show that such stochastic equations arise within the modern paradigm of derivative pricing where a central counterparty (CCP) demands each member to deposit initial and variation margins to cover their exp...
Agarwal, Ankush; De Marco, Stefano; GOBET, Emmanuel; López-Salas, José,; Noubiagain, Fanny; Zhou, Alexandre
2018-01-01
We introduce a new class of anticipative backward stochastic differential equations with a dependence of McKean type on the law of the solution, that we name MKABSDE. We provide existence and uniqueness results in a general framework with relaxed regularity assumptions on the parameters. We show that such stochastic equations arise within the modern paradigm of derivative pricing where a central counterparty (CCP) demands each member to deposit variation and initial margins to cover their exp...
Directory of Open Access Journals (Sweden)
Hua Yang
2012-01-01
Full Text Available We are concerned with the stochastic differential delay equations with Poisson jump and Markovian switching (SDDEsPJMSs. Most SDDEsPJMSs cannot be solved explicitly as stochastic differential equations. Therefore, numerical solutions have become an important issue in the study of SDDEsPJMSs. The key contribution of this paper is to investigate the strong convergence between the true solutions and the numerical solutions to SDDEsPJMSs when the drift and diffusion coefficients are Taylor approximations.
Ergodicity for a Stochastic Geodesic Equation in the Tangent Bundle of the 2D Sphere
Czech Academy of Sciences Publication Activity Database
Baňas, L.; Brzezniak, Z.; Neklyudov, M.; Ondreját, Martin; Prohl, A.
2015-01-01
Roč. 65, č. 3 (2015), s. 617-657 ISSN 0011-4642 R&D Projects: GA ČR GAP201/10/0752 Institutional support: RVO:67985556 Keywords : geometric stochastic wave equation * stochastic geodesic equation * ergodicity * attractivity * invariant measure * numerical approximation Subject RIV: BA - General Mathematics Impact factor: 0.284, year: 2015 http://library.utia.cas.cz/separaty/2015/SI/ondrejat-0451399.pdf
Existence and uniqueness of solution for a class of stochastic differential equations.
Cao, Junfei; Huang, Zaitang; Zeng, Caibin
2013-01-01
A class of stochastic differential equations given by dx(t) = f(x(t))dt + g(x(t))dW(t), x(t 0) = x 0, t 0 ≤ t ≤ T equations are obtained. Moreover, the existence and uniqueness of solution for stochastic Lorenz system, which is illustrated by example, are in good agreement with the theoretical analysis.
DEFF Research Database (Denmark)
Tornøe, Christoffer Wenzel; Overgaard, Rune Viig; Agerso, H.
2005-01-01
of noise: a measurement and a system noise term. The measurement noise represents uncorrelated error due to, for example, assay error while the system noise accounts for structural misspecifications, approximations of the dynamical model, and true random physiological fluctuations. Since the system noise......Purpose. The objective of the present analysis was to explore the use of stochastic differential equations (SDEs) in population pharmacokinetic/pharmacodynamic (PK/PD) modeling. Methods. The intra-individual variability in nonlinear mixed-effects models based on SDEs is decomposed into two types...... accounts for model misspecifications, the SDEs provide a diagnostic tool for model appropriateness. The focus of the article is on the implementation of the Extended Kalman Filter (EKF) in NONMEM(R) for parameter estimation in SDE models. Results. Various applications of SDEs in population PK/PD modeling...
Directory of Open Access Journals (Sweden)
Mourad Kerboua
2014-12-01
Full Text Available We introduce a new notion called fractional stochastic nonlocal condition, and then we study approximate controllability of class of fractional stochastic nonlinear differential equations of Sobolev type in Hilbert spaces. We use Hölder's inequality, fixed point technique, fractional calculus, stochastic analysis and methods adopted directly from deterministic control problems for the main results. A new set of sufficient conditions is formulated and proved for the fractional stochastic control system to be approximately controllable. An example is given to illustrate the abstract results.
Numerical simulation of stochastic point kinetic equation in the dynamical system of nuclear reactor
International Nuclear Information System (INIS)
Saha Ray, S.
2012-01-01
Highlights: ► In this paper stochastic neutron point kinetic equations have been analyzed. ► Euler–Maruyama method and Strong Taylor 1.5 order method have been discussed. ► These methods are applied for the solution of stochastic point kinetic equations. ► Comparison between the results of these methods and others are presented in tables. ► Graphs for neutron and precursor sample paths are also presented. -- Abstract: In the present paper, the numerical approximation methods, applied to efficiently calculate the solution for stochastic point kinetic equations () in nuclear reactor dynamics, are investigated. A system of Itô stochastic differential equations has been analyzed to model the neutron density and the delayed neutron precursors in a point nuclear reactor. The resulting system of Itô stochastic differential equations are solved over each time-step size. The methods are verified by considering different initial conditions, experimental data and over constant reactivities. The computational results indicate that the methods are simple and suitable for solving stochastic point kinetic equations. In this article, a numerical investigation is made in order to observe the random oscillations in neutron and precursor population dynamics in subcritical and critical reactors.
A Stability Result for Stochastic Differential Equations Driven by Fractional Brownian Motions
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Bruno Saussereau
2012-01-01
Full Text Available We study the stability of the solutions of stochastic differential equations driven by fractional Brownian motions with Hurst parameter greater than half. We prove that when the initial conditions, the drift, and the diffusion coefficients as well as the fractional Brownian motions converge in a suitable sense, then the sequence of the solutions of the corresponding equations converge in Hölder norm to the solution of a stochastic differential equation. The limit equation is driven by the limit fractional Brownian motion and its coefficients are the limits of the sequence of the coefficients.
Cotter, C J; Gottwald, G A; Holm, D D
2017-09-01
In Holm (Holm 2015 Proc. R. Soc. A 471 , 20140963. (doi:10.1098/rspa.2014.0963)), stochastic fluid equations were derived by employing a variational principle with an assumed stochastic Lagrangian particle dynamics. Here we show that the same stochastic Lagrangian dynamics naturally arises in a multi-scale decomposition of the deterministic Lagrangian flow map into a slow large-scale mean and a rapidly fluctuating small-scale map. We employ homogenization theory to derive effective slow stochastic particle dynamics for the resolved mean part, thereby obtaining stochastic fluid partial equations in the Eulerian formulation. To justify the application of rigorous homogenization theory, we assume mildly chaotic fast small-scale dynamics, as well as a centring condition. The latter requires that the mean of the fluctuating deviations is small, when pulled back to the mean flow.
Bessaih, Hakima
2015-04-01
The evolution Stokes equation in a domain containing periodically distributed obstacles subject to Fourier boundary condition on the boundaries is considered. We assume that the dynamic is driven by a stochastic perturbation on the interior of the domain and another stochastic perturbation on the boundaries of the obstacles. We represent the solid obstacles by holes in the fluid domain. The macroscopic (homogenized) equation is derived as another stochastic partial differential equation, defined in the whole non perforated domain. Here, the initial stochastic perturbation on the boundary becomes part of the homogenized equation as another stochastic force. We use the twoscale convergence method after extending the solution with 0 in the holes to pass to the limit. By Itô stochastic calculus, we get uniform estimates on the solution in appropriate spaces. In order to pass to the limit on the boundary integrals, we rewrite them in terms of integrals in the whole domain. In particular, for the stochastic integral on the boundary, we combine the previous idea of rewriting it on the whole domain with the assumption that the Brownian motion is of trace class. Due to the particular boundary condition dealt with, we get that the solution of the stochastic homogenized equation is not divergence free. However, it is coupled with the cell problem that has a divergence free solution. This paper represents an extension of the results of Duan and Wang (Comm. Math. Phys. 275:1508-1527, 2007), where a reaction diffusion equation with a dynamical boundary condition with a noise source term on both the interior of the domain and on the boundary was studied, and through a tightness argument and a pointwise two scale convergence method the homogenized equation was derived. © American Institute of Mathematical Sciences.
Volkova, Victoriya V; Lu, Zhao; Lanzas, Cristina; Scott, H Morgan; Gröhn, Yrjö T
2013-01-01
The ubiquitous commensal bacteria harbour genes of antimicrobial resistance (AMR), often on conjugative plasmids. Antimicrobial use in food animals subjects their enteric commensals to antimicrobial pressure. A fraction of enteric Escherichia coli in cattle exhibit plasmid-gene mediated AMR to a third-generation cephalosporin ceftiofur. We adapted stochastic differential equations with diffusion approximation (a compartmental stochastic mathematical model) to research the sources and roles of stochasticity in the resistance dynamics, both during parenteral antimicrobial therapy and in its absence. The results demonstrated that demographic stochasticity among enteric E. coli in the occurrence of relevant events was important for the AMR dynamics only when bacterial numbers were depressed during therapy. However, stochasticity in the parameters of enteric E. coli ecology, whether externally or intrinsically driven, contributed to a wider distribution of the resistant E. coli fraction, both during therapy and in its absence, with stochasticities in individual parameters interacting in their contribution.
Nonperturbative renormalization group study of the stochastic Navier-Stokes equation.
Mejía-Monasterio, Carlos; Muratore-Ginanneschi, Paolo
2012-07-01
We study the renormalization group flow of the average action of the stochastic Navier-Stokes equation with power-law forcing. Using Galilean invariance, we introduce a nonperturbative approximation adapted to the zero-frequency sector of the theory in the parametric range of the Hölder exponent 4-2ε of the forcing where real-space local interactions are relevant. In any spatial dimension d, we observe the convergence of the resulting renormalization group flow to a unique fixed point which yields a kinetic energy spectrum scaling in agreement with canonical dimension analysis. Kolmogorov's -5/3 law is, thus, recovered for ε = 2 as also predicted by perturbative renormalization. At variance with the perturbative prediction, the -5/3 law emerges in the presence of a saturation in the ε dependence of the scaling dimension of the eddy diffusivity at ε = 3/2 when, according to perturbative renormalization, the velocity field becomes infrared relevant.
A model for closing the inviscid form of the average-passage equation system
Adamczyk, J. J.; Mulac, R. A.; Celestina, M. L.
1986-01-01
A mathematical model is proposed for closing or mathematically completing the system of equations which describes the time average flow field through the blade passages of multistage turbomachinery. These equations referred to as the average passage equation system govern a conceptual model which has proven useful in turbomachinery aerodynamic design and analysis. The closure model is developed so as to insure a consistency between these equations and the axisymmetric through flow equations. The closure model was incorporated into a computer code for use in simulating the flow field about a high speed counter rotating propeller and a high speed fan stage. Results from these simulations are presented.
Dynamically orthogonal field equations for stochastic flows and particle dynamics
2011-02-01
fields implies orthogonality of their spatial Fourier, Gabor, and Wavelet transforms [5], [35]. Therefore, different DO modes always contain different...modes stochastic solution u (x, t;ω) to the stochastic subspace computed using the five modes solution, i.e. σ2i (t) = var [〈u (x, t;ω) ,vi (x, t...Antoine, R. Murenzi, P. Vandergheynst, and S.T. Ali. Two-dimensional wavelets and their relatives. Cambridge University Press, 2004. [6] V.I. Arnold and
White noise solutions to the stochastic mKdV equation
International Nuclear Information System (INIS)
Zhang Zhongjun; Wei Caimin
2009-01-01
In this paper, we present the white noise solutions of the stochastic mKdV equation via the Hermite transformation and variable-coefficient generalized projected Ricatti equation expansion method. These solutions include white noise solitary wave solutions, white noise soliton-like solutions and white noise trigonometric function solutions.
Directory of Open Access Journals (Sweden)
Qiong Wu
2015-01-01
Full Text Available Mild solutions generated by a a, k-regularized family to fractional stochastic relaxation equations are studied. The main objective is to establish the existence and uniqueness of square-mean asymptotically almost automorphic mild solutions to linear and semilinear case of these equations. Under different hypotheses, some new theorems concerning the main objective are derived.
A stochastic differential equation framework for the timewise dynamics of turbulent velocities
DEFF Research Database (Denmark)
Barndorff-Nielsen, Ole Eiler; Schmiegel, Jürgen
2008-01-01
We discuss a stochastic differential equation as a modeling framework for the timewise dynamics of turbulent velocities. The equation is capable of capturing basic stylized facts of the statistics of temporal velocity increments. In particular, we focus on the evolution of the probability density...
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N. Ahmadi
2017-02-01
Full Text Available Abstract In this paper, we present a collocation method based on Gaussian Radial Basis Functions (RBFs for approximating the solution of stochastic fractional differential equations (SFDEs. In this equation the fractional derivative is considered in the Caputo sense. Also we prove the existence and uniqueness of the presented method. Numerical examples confirm the proficiency of the method.
International Nuclear Information System (INIS)
Zhang, Zhongqiang; Yang, Xiu; Lin, Guang; Karniadakis, George Em
2013-01-01
We consider a piston with a velocity perturbed by Brownian motion moving into a straight tube filled with a perfect gas at rest. The shock generated ahead of the piston can be located by solving the one-dimensional Euler equations driven by white noise using the Stratonovich or Ito formulations. We approximate the Brownian motion with its spectral truncation and subsequently apply stochastic collocation using either sparse grid or the quasi-Monte Carlo (QMC) method. In particular, we first transform the Euler equations with an unsteady stochastic boundary into stochastic Euler equations over a fixed domain with a time-dependent stochastic source term. We then solve the transformed equations by splitting them up into two parts, i.e., a ‘deterministic part’ and a ‘stochastic part’. Numerical results verify the Stratonovich–Euler and Ito–Euler models against stochastic perturbation results, and demonstrate the efficiency of sparse grid and QMC for small and large random piston motions, respectively. The variance of shock location of the piston grows cubically in the case of white noise in contrast to colored noise reported in [1], where the variance of shock location grows quadratically with time for short times and linearly for longer times
Heat kernel regularization of the effective action for stochastic reaction-diffusion equations.
Hochberg, D; Molina-París, C; Visser, M
2001-03-01
The presence of fluctuations and nonlinear interactions can lead to scale dependence in the parameters appearing in stochastic differential equations. Stochastic dynamics can be formulated in terms of functional integrals. In this paper we apply the heat kernel method to study the short distance renormalizability of a stochastic (polynomial) reaction-diffusion equation with real additive noise. We calculate the one-loop effective action and its ultraviolet scale dependent divergences. We show that for white noise a polynomial reaction-diffusion equation is one-loop finite in d=0 and d=1, and is one-loop renormalizable in d=2 and d=3 space dimensions. We obtain the one-loop renormalization group equations and find they run with scale only in d=2.
International Nuclear Information System (INIS)
Grigoriu, Mircea; Samorodnitsky, Gennady
2004-01-01
Two methods are considered for assessing the asymptotic stability of the trivial solution of linear stochastic differential equations driven by Poisson white noise, interpreted as the formal derivative of a compound Poisson process. The first method attempts to extend a result for diffusion processes satisfying linear stochastic differential equations to the case of linear equations with Poisson white noise. The developments for the method are based on Ito's formula for semimartingales and Lyapunov exponents. The second method is based on a geometric ergodic theorem for Markov chains providing a criterion for the asymptotic stability of the solution of linear stochastic differential equations with Poisson white noise. Two examples are presented to illustrate the use and evaluate the potential of the two methods. The examples demonstrate limitations of the first method and the generality of the second method
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Eman Ali Hussain
2015-01-01
Full Text Available Absract In this project A new method for solving Stochastic Differential Equations SDEs deriving by Wiener process numerically will be construct and implement using Accelerated Genetic Algorithm AGA. An SDE is a differential equation in which one or more of the terms and hence the solutions itself is a stochastic process. Solving stochastic differential equations requires going away from the recognizable deterministic setting of ordinary and partial differential equations into a world where the evolution of a quantity has an inherent random component and where the expected behavior of this quantity can be described in terms of probability distributions. We applied our method on the Ito formula which is equivalent to the SDE to find approximation solution of the SDEs. Numerical experiments illustrate the behavior of the proposed method.
Large deviations for solutions to stochastic recurrence equations under Kesten's condition
DEFF Research Database (Denmark)
Buraczewski, Dariusz; Damek, Ewa; Mikosch, Thomas Valentin
2013-01-01
In this paper we prove large deviations results for partial sums constructed from the solution to a stochastic recurrence equation. We assume Kesten’s condition [17] under which the solution of the stochastic recurrence equation has a marginal distribution with power law tails, while the noise...... sequence of the equations can have light tails. The results of the paper are analogs of those obtained by A.V. and S.V. Nagaev [21, 22] in the case of partial sums of iid random variables. In the latter case, the large deviation probabilities of the partial sums are essentially determined by the largest...... step size of the partial sum. For the solution to a stochastic recurrence equation, the magnitude of the large deviation probabilities is again given by the tail of the maximum summand, but the exact asymptotic tail behavior is also influenced by clusters of extreme values, due to dependencies...
Nonexplosion of solutions to stochastic reaction-diffusion equations
Czech Academy of Sciences Publication Activity Database
Dozzi, M.; Maslowski, Bohdan
2002-01-01
Roč. 82, - (2002), s. 745-751 ISSN 0044-2267 Institutional research plan: CEZ:AV0Z1019905 Keywords : stochastic parabolic PDEďs%nonexplosion Subject RIV: ba - General Mathematics Impact factor: 0.085, year: 2002
A Hierarchical Latent Stochastic Differential Equation Model for Affective Dynamics
Oravecz, Zita; Tuerlinckx, Francis; Vandekerckhove, Joachim
2011-01-01
In this article a continuous-time stochastic model (the Ornstein-Uhlenbeck process) is presented to model the perpetually altering states of the core affect, which is a 2-dimensional concept underlying all our affective experiences. The process model that we propose can account for the temporal changes in core affect on the latent level. The key…
Limit theorems for solutions of stochastic differential equation problems
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J. Vom Scheidt
1980-01-01
Full Text Available In this paper linear differential equations with random processes as coefficients and as inhomogeneous term are regarded. Limit theorems are proved for the solutions of these equations if the random processes are weakly correlated processes.
Lv, Qiming; Schneider, Manuel K; Pitchford, Jonathan W
2008-08-01
We study individual plant growth and size hierarchy formation in an experimental population of Arabidopsis thaliana, within an integrated analysis that explicitly accounts for size-dependent growth, size- and space-dependent competition, and environmental stochasticity. It is shown that a Gompertz-type stochastic differential equation (SDE) model, involving asymmetric competition kernels and a stochastic term which decreases with the logarithm of plant weight, efficiently describes individual plant growth, competition, and variability in the studied population. The model is evaluated within a Bayesian framework and compared to its deterministic counterpart, and to several simplified stochastic models, using distributional validation. We show that stochasticity is an important determinant of size hierarchy and that SDE models outperform the deterministic model if and only if structural components of competition (asymmetry; size- and space-dependence) are accounted for. Implications of these results are discussed in the context of plant ecology and in more general modelling situations.
Derivation of stochastic partial differential equations for size- and age-structured populations.
Allen, Edward J
2009-01-01
Stochastic partial differential equations (SPDEs) for size-structured and age- and size-structured populations are derived from basic principles, i.e. from the changes that occur in a small time interval. Discrete stochastic models of size-structured and age-structured populations are constructed, carefully taking into account the inherent randomness in births, deaths, and size changes. As the time interval decreases, the discrete stochastic models lead to systems of Itô stochastic differential equations. As the size and age intervals decrease, SPDEs are derived for size-structured and age- and size-structured populations. Comparisons between numerical solutions of the SPDEs and independently formulated Monte Carlo calculations support the accuracy of the derivations.
Tornøe, Christoffer W; Overgaard, Rune V; Agersø, Henrik; Nielsen, Henrik A; Madsen, Henrik; Jonsson, E Niclas
2005-08-01
The objective of the present analysis was to explore the use of stochastic differential equations (SDEs) in population pharmacokinetic/pharmacodynamic (PK/PD) modeling. The intra-individual variability in nonlinear mixed-effects models based on SDEs is decomposed into two types of noise: a measurement and a system noise term. The measurement noise represents uncorrelated error due to, for example, assay error while the system noise accounts for structural misspecifications, approximations of the dynamical model, and true random physiological fluctuations. Since the system noise accounts for model misspecifications, the SDEs provide a diagnostic tool for model appropriateness. The focus of the article is on the implementation of the Extended Kalman Filter (EKF) in NONMEM for parameter estimation in SDE models. Various applications of SDEs in population PK/PD modeling are illustrated through a systematic model development example using clinical PK data of the gonadotropin releasing hormone (GnRH) antagonist degarelix. The dynamic noise estimates were used to track variations in model parameters and systematically build an absorption model for subcutaneously administered degarelix. The EKF-based algorithm was successfully implemented in NONMEM for parameter estimation in population PK/PD models described by systems of SDEs. The example indicated that it was possible to pinpoint structural model deficiencies, and that valuable information may be obtained by tracking unexplained variations in parameters.
A model for closing the inviscid form of the average passage equation system
Adamczyk, John J.; Mulac, R. A.; Celestina, M. L.
1996-01-01
A mathematical model for closing or mathematically completing the system of equations is proposed. The model describes the time average flow field through the blade passages of multistage turbomachinery. These average-passage equation systems govern a conceptual model useful in turbomachinery aerodynamic design and analysis. The closure model was developed to insure a consistency between these equations and the axisymmetric through-flow equations. The closure model was incorporated into a calculation code for use in the simulation of the flow field about a high-speed counter rotating propeller and a high-speed fan stage.
Stochastic Effects for the Reaction-Duffing Equation with Wick-Type Product
Directory of Open Access Journals (Sweden)
Jin Hyuk Choi
2016-01-01
Full Text Available We construct new explicit solutions of the Wick-type stochastic reaction-Duffing equation arising from mathematical physics with the help of the white noise theory and the system technique. Based on these exact solutions, we also discuss the influences of stochastic effects for dynamical behaviors according to functions h1(t, h2(t, and Brownian motion B(t which are the solitary wave group velocities.
Existence and large time behavior for a stochastic model of modified magnetohydrodynamic equations
Razafimandimby, Paul André; Sango, Mamadou
2015-10-01
In this paper, we study a system of nonlinear stochastic partial differential equations describing the motion of turbulent non-Newtonian media in the presence of fluctuating magnetic field. The system is basically obtained by a coupling of the dynamical equations of a non-Newtonian fluids having p-structure and the Maxwell equations. We mainly show the existence of weak martingale solutions and their exponential decay when time goes to infinity.
DEFF Research Database (Denmark)
Duun-Henriksen, Anne Katrine; Schmidt, S.; Nørgaard, K.
2013-01-01
extension incorporating exercise effects on insulin and glucose dynamics. Our model is constructed as a stochastic state space model consisting of a set of stochastic differential equations (SDEs). In a stochastic state space model, the residual error is split into random measurement error...... physical activity. Exercise constitutes a substantial challenge to closed-loop control of T1D. The effects are many and depend on intensity and duration and may be delayed by several hours. In this study, we use a model for the glucoregulatory system based on the minimal model and a previously published...
Fast Numerical Methods for Stochastic Partial Differential Equations
2016-04-15
applicable SPDES with efficient numerical methods . This project is intended to address the numerical analysis as well as algorithm aspects of SPDES. Three...no. 1, 784–804. 10. Cao, Yanzhao; Wang, Peng; Wang, Xiaoshen Homotopy continuation methods for stochastic two-point boundary value problems driven by...convergence analysis of Quasi Monte Carlo based Particle Swarm Optimization (PSO) method ; ii) Efficient adaptive domain sparse grid method for SPDES; iii
Set-valued and fuzzy stochastic integral equations driven by semimartingales under Osgood condition
Directory of Open Access Journals (Sweden)
Malinowski Marek T.
2015-01-01
Full Text Available We analyze the set-valued stochastic integral equations driven by continuous semimartingales and prove the existence and uniqueness of solutions to such equations in the framework of the hyperspace of nonempty, bounded, convex and closed subsets of the Hilbert space L2 (consisting of square integrable random vectors. The coefficients of the equations are assumed to satisfy the Osgood type condition that is a generalization of the Lipschitz condition. Continuous dependence of solutions with respect to data of the equation is also presented. We consider equations driven by semimartingale Z and equations driven by processes A;M from decomposition of Z, where A is a process of finite variation and M is a local martingale. These equations are not equivalent. Finally, we show that the analysis of the set-valued stochastic integral equations can be extended to a case of fuzzy stochastic integral equations driven by semimartingales under Osgood type condition. To obtain our results we use the set-valued and fuzzy Maruyama type approximations and Bihari’s inequality.
Short-term Probabilistic Forecasting of Wind Speed Using Stochastic Differential Equations
DEFF Research Database (Denmark)
Iversen, Jan Emil Banning; Morales González, Juan Miguel; Møller, Jan Kloppenborg
2016-01-01
It is widely accepted today that probabilistic forecasts of wind power production constitute valuable information for both wind power producers and power system operators to economically exploit this form of renewable energy, while mitigating the potential adverse effects related to its variable...... and uncertain nature. In this paper, we propose a modeling framework for wind speed that is based on stochastic differential equations. We show that stochastic differential equations allow us to naturally capture the time dependence structure of wind speed prediction errors (from 1 up to 24 hours ahead) and......, most importantly, to derive point and quantile forecasts, predictive distributions, and time-path trajectories (also referred to as scenarios or ensemble forecasts), all by one single stochastic differential equation model characterized by a few parameters....
McCarthy, S.; Rachinskii, D.
2011-01-01
We describe two Euler type numerical schemes obtained by discretisation of a stochastic differential equation which contains the Preisach memory operator. Equations of this type are of interest in areas such as macroeconomics and terrestrial hydrology where deterministic models containing the Preisach operator have been developed but do not fully encapsulate stochastic aspects of the area. A simple price dynamics model is presented as one motivating example for our studies. Some numerical evidence is given that the two numerical schemes converge to the same limit as the time step decreases. We show that the Preisach term introduces a damping effect which increases on the parts of the trajectory demonstrating a stronger upwards or downwards trend. The results are preliminary to a broader programme of research of stochastic differential equations with the Preisach hysteresis operator.
Numerical Integration of the Master Equation in Some Models of Stochastic Epidemiology
Jenkinson, Garrett; Goutsias, John
2012-01-01
The processes by which disease spreads in a population of individuals are inherently stochastic. The master equation has proven to be a useful tool for modeling such processes. Unfortunately, solving the master equation analytically is possible only in limited cases (e.g., when the model is linear), and thus numerical procedures or approximation methods must be employed. Available approximation methods, such as the system size expansion method of van Kampen, may fail to provide reliable solutions, whereas current numerical approaches can induce appreciable computational cost. In this paper, we propose a new numerical technique for solving the master equation. Our method is based on a more informative stochastic process than the population process commonly used in the literature. By exploiting the structure of the master equation governing this process, we develop a novel technique for calculating the exact solution of the master equation – up to a desired precision – in certain models of stochastic epidemiology. We demonstrate the potential of our method by solving the master equation associated with the stochastic SIR epidemic model. MATLAB software that implements the methods discussed in this paper is freely available as Supporting Information S1. PMID:22615755
Derivation of a volume-averaged neutron diffusion equation; Atomos para el desarrollo de Mexico
Energy Technology Data Exchange (ETDEWEB)
Vazquez R, R.; Espinosa P, G. [UAM-Iztapalapa, Av. San Rafael Atlixco 186, Col. Vicentina, Mexico D.F. 09340 (Mexico); Morales S, Jaime B. [UNAM, Laboratorio de Analisis en Ingenieria de Reactores Nucleares, Paseo Cuauhnahuac 8532, Jiutepec, Morelos 62550 (Mexico)]. e-mail: rvr@xanum.uam.mx
2008-07-01
This paper presents a general theoretical analysis of the problem of neutron motion in a nuclear reactor, where large variations on neutron cross sections normally preclude the use of the classical neutron diffusion equation. A volume-averaged neutron diffusion equation is derived which includes correction terms to diffusion and nuclear reaction effects. A method is presented to determine closure-relationships for the volume-averaged neutron diffusion equation (e.g., effective neutron diffusivity). In order to describe the distribution of neutrons in a highly heterogeneous configuration, it was necessary to extend the classical neutron diffusion equation. Thus, the volume averaged diffusion equation include two corrections factor: the first correction is related with the absorption process of the neutron and the second correction is a contribution to the neutron diffusion, both parameters are related to neutron effects on the interface of a heterogeneous configuration. (Author)
On a stochastic Burgers equation with Dirichlet boundary conditions
Directory of Open Access Journals (Sweden)
Ekaterina T. Kolkovska
2003-01-01
Full Text Available We consider the one-dimensional Burgers equation perturbed by a white noise term with Dirichlet boundary conditions and a non-Lipschitz coefficient. We obtain existence of a weak solution proving tightness for a sequence of polygonal approximations for the equation and solving a martingale problem for the weak limit.
Weak Second Order Explicit Stabilized Methods for Stiff Stochastic Differential Equations
Abdulle, Assyr
2013-01-01
We introduce a new family of explicit integrators for stiff Itô stochastic differential equations (SDEs) of weak order two. These numerical methods belong to the class of one-step stabilized methods with extended stability domains and do not suffer from the step size reduction faced by standard explicit methods. The family is based on the standard second order orthogonal Runge-Kutta-Chebyshev (ROCK2) methods for deterministic problems. The convergence, meansquare, and asymptotic stability properties of the methods are analyzed. Numerical experiments, including applications to nonlinear SDEs and parabolic stochastic partial differential equations are presented and confirm the theoretical results. © 2013 Society for Industrial and Applied Mathematics.
Directory of Open Access Journals (Sweden)
Minghui Song
2012-01-01
Full Text Available The main purpose of this paper is to investigate the convergence of the Euler method to stochastic differential equations with piecewise continuous arguments (SEPCAs. The classical Khasminskii-type theorem gives a powerful tool to examine the global existence of solutions for stochastic differential equations (SDEs without the linear growth condition by the use of the Lyapunov functions. However, there is no such result for SEPCAs. Firstly, this paper shows SEPCAs which have nonexplosion global solutions under local Lipschitz condition without the linear growth condition. Then the convergence in probability of numerical solutions to SEPCAs under the same conditions is established. Finally, an example is provided to illustrate our theory.
Averaging of the Equations of the Standard Cosmological Model over Rapid Oscillations
Ignat'ev, Yu. G.; Samigullina, A. R.
2017-11-01
An averaging of the equations of the standard cosmological model (SCM) is carried out. It is shown that the main contribution to the macroscopic energy density of the scalar field comes from its microscopic oscillations with the Compton period. The effective macroscopic equation of state of the oscillations of the scalar field corresponds to the nonrelativistic limit.
Modelling of nonlinear shoaling based on stochastic evolution equations
DEFF Research Database (Denmark)
Kofoed-Hansen, Henrik; Rasmussen, Jørgen Hvenekær
1998-01-01
are recast into evolution equations for the complex amplitudes, and serve as the underlying deterministic model. Next, a set of evolution equations for the cumulants is derived. By formally introducing the well-known Gaussian closure hypothesis, nonlinear evolution equations for the power spectrum...... with experimental data in four different cases as well as with the underlying deterministic model. In general, the agreement is found to be acceptable, even far beyond the region where Gaussianity (Gaussian sea state) may be justified. (C) 1998 Elsevier Science B.V....
Lyapunov functionals and stability of stochastic difference equations
Shaikhet, Leonid
2011-01-01
This book offers a general method of Lyapunov functional construction which lets researchers analyze the degree to which the stability properties of differential equations are preserved in their difference analogues. Includes examples from physical systems.
Klyatskin, Valery I
2015-01-01
This monograph set presents a consistent and self-contained framework of stochastic dynamic systems with maximal possible completeness. Volume 1 presents the basic concepts, exact results, and asymptotic approximations of the theory of stochastic equations on the basis of the developed functional approach. This approach offers a possibility of both obtaining exact solutions to stochastic problems for a number of models of fluctuating parameters and constructing various asymptotic buildings. Ideas of statistical topography are used to discuss general issues of generating coherent structures from chaos with probability one, i.e., almost in every individual realization of random parameters. The general theory is illustrated with certain problems and applications of stochastic mathematical physics in various fields such as mechanics, hydrodynamics, magnetohydrodynamics, acoustics, optics, and radiophysics.
Krylov, N. V.; Priola, E.
2017-09-01
We show, among other things, how knowing Schauder or Sobolev-space estimates for the one-dimensional heat equation allows one to derive their multidimensional analogs for equations with coefficients depending only on the time variable with the same constants as in the case of the one-dimensional heat equation. The method is quite general and is based on using the Poisson stochastic process. It also applies to equations involving non-local operators. It looks like no other methods are available at this time and it is a very challenging problem to find a purely analytical approach to proving such results.
Directory of Open Access Journals (Sweden)
Diem Dang Huan
2015-12-01
Full Text Available The current paper is concerned with the controllability of nonlocal second-order impulsive neutral stochastic functional integro-differential equations with infinite delay and Poisson jumps in Hilbert spaces. Using the theory of a strongly continuous cosine family of bounded linear operators, stochastic analysis theory and with the help of the Banach fixed point theorem, we derive a new set of sufficient conditions for the controllability of nonlocal second-order impulsive neutral stochastic functional integro-differential equations with infinite delay and Poisson jumps. Finally, an application to the stochastic nonlinear wave equation with infinite delay and Poisson jumps is given.
International Nuclear Information System (INIS)
Levenshtam, V B
2006-01-01
We justify the averaging method for abstract parabolic equations with stationary principal part that contain non-linearities (subordinate to the principal part) some of whose terms are rapidly oscillating in time with zero mean and are proportional to the square root of the frequency of oscillation. Our interest in the exponent 1/2 is motivated by the fact that terms proportional to lower powers of the frequency have no influence on the average. For linear equations of the same type, we justify an algorithm for the study of the stability of solutions in the case when the stationary averaged problem has eigenvalues on the imaginary axis (the critical case)
Stochastic differential equation models for ion channel noise in Hodgkin-Huxley neurons
Goldwyn, Joshua H.; Imennov, Nikita S.; Famulare, Michael; Shea-Brown, Eric
2011-01-01
The random transitions of ion channels between conducting and nonconducting states generate a source of internal fluctuations in a neuron, known as channel noise. The standard method for modeling the states of ion channels nonlinearly couples continuous-time Markov chains to a differential equation for voltage. Beginning with the work of R. F. Fox and Y.-N. Lu [Phys. Rev. E 49, 3421 (1994)], there have been attempts to generate simpler models that use stochastic differential equation (SDEs) t...
International Nuclear Information System (INIS)
Bouard, Anne de; Debussche, Arnaud
2006-01-01
In this article we analyze the error of a semidiscrete scheme for the stochastic nonlinear Schrodinger equation with power nonlinearity. We consider supercritical or subcritical nonlinearity and the equation can be either focusing or defocusing. Allowing sufficient spatial regularity we prove that the numerical scheme has strong order 1/2 in general and order 1 if the noise is additive. Furthermore, we also prove that the weak order is always 1
Parameter estimation in a simple stochastic differential equation for phytoplankton modelling
DEFF Research Database (Denmark)
Møller, Jan Kloppenborg; Madsen, Henrik; Carstensen, Jacob
2011-01-01
The use of stochastic differential equations (SDEs) for simulation of aquatic ecosystems has attracted increasing attention in recent years. The SDE setting also provides the opportunity for statistical estimation of ecosystem parameters. We present an estimation procedure, based on Kalman...... filtering and likelihood estimation, which has proven useful in other fields of application. The estimation procedure is presented and the development from ordinary differential equations (ODEs) to SDEs is discussed with emphasis on autocorrelated residuals, commonly encountered with ODEs. The estimation...
Weak Solutions to Stochastic Wave Equations with Values in Riemannian Manifolds
Czech Academy of Sciences Publication Activity Database
Brzezniak, Z.; Ondreját, Martin
2011-01-01
Roč. 36, č. 9 (2011), s. 1624-1653 ISSN 0360-5302 R&D Projects: GA ČR GA201/07/0237 Institutional research plan: CEZ:AV0Z10750506 Keywords : geometric wave equation * stochastic wave equation Subject RIV: BA - General Mathematics Impact factor: 0.894, year: 2011 http://library.utia.cas.cz/separaty/2011/SI/ondrejat-0362936.pdf
An auto-Baecklund transformation and exact solutions of stochastic Wick-type Sawada-Kotera equations
Energy Technology Data Exchange (ETDEWEB)
Chen Bin E-mail: journal@xznu.edu.cn; Xie Yingchao E-mail: ycxie588@public.xz.js.cn
2005-01-01
This paper shows an auto-Baecklund transformation and soliton solutions for variable coefficient Sawada-Kotera equations and stochastic soliton solutions of stochastic Wick-type Sawada-Kotera equations by using the Hermite transform in Kondratiev distribution space (S){sub -1}.
Transport modelling in coastal waters using stochastic differential equations
Charles, W.M.
2007-01-01
In this thesis, the particle model that takes into account the short term correlation behaviour of pollutants dispersion has been developed. An efficient particle model for sediment transport has been developed. We have modified the existing particle model by adding extra equations for the
Advances in nonlinear partial differential equations and stochastics
Kawashima, S
1998-01-01
In the past two decades, there has been great progress in the theory of nonlinear partial differential equations. This book describes the progress, focusing on interesting topics in gas dynamics, fluid dynamics, elastodynamics etc. It contains ten articles, each of which discusses a very recent result obtained by the author. Some of these articles review related results.
Directory of Open Access Journals (Sweden)
John A. D. Appleby
2008-01-01
Full Text Available This paper considers necessary and sufficient conditions for the solution of a stochastically and deterministically perturbed Volterra equation to converge exponentially to a nonequilibrium and nontrivial limit. Convergence in an almost sure and pth mean sense is obtained.
Energy Technology Data Exchange (ETDEWEB)
Cui Jing [College of Information Science and Technology, Donghua University, 2999 North Renmin Rd, Songjiang, Shanghai 201620 (China); Yan Litan, E-mail: jcui123@126.com, E-mail: litanyan@hotmail.com [Department of Mathematics, College of Science, Donghua University, 2999 North Renmin Rd, Songjiang, Shanghai 201620 (China)
2011-08-19
This paper is concerned with the existence of mild solutions for a class of fractional neutral stochastic integro-differential equations with infinite delay in Hilbert spaces. A sufficient condition for the existence is obtained under non-Lipschitz conditions by means of Sadovskii's fixed point theorem. An example is given to illustrate the theory.
Picard Approximation of Stochastic Differential Equations and Application to LIBOR Models
DEFF Research Database (Denmark)
Papapantoleon, Antonis; Skovmand, David
The aim of this work is to provide fast and accurate approximation schemes for the Monte Carlo pricing of derivatives in LIBOR market models. Standard methods can be applied to solve the stochastic differential equations of the successive LIBOR rates but the methods are generally slow. Our...
Large Deviations for Stochastic Tamed 3D Navier-Stokes Equations
International Nuclear Information System (INIS)
Roeckner, Michael; Zhang, Tusheng; Zhang Xicheng
2010-01-01
In this paper, using weak convergence method, we prove a large deviation principle of Freidlin-Wentzell type for the stochastic tamed 3D Navier-Stokes equations driven by multiplicative noise, which was investigated in (Roeckner and Zhang in Probab. Theory Relat. Fields 145(1-2), 211-267, 2009).
International Nuclear Information System (INIS)
Frank, T D
2005-01-01
Stationary distributions of processes are derived that involve a time delay and are defined by a linear stochastic neutral delay differential equation. The distributions are Gaussian distributions. The variances of the Gaussian distributions are either monotonically increasing or decreasing functions of the time delays. The variances become infinite when fixed points of corresponding deterministic processes become unstable. (letter to the editor)
DEFF Research Database (Denmark)
Møller, Jan Kloppenborg; Philipsen, Kirsten Riber; Christiansen, Lasse Engbo
2012-01-01
In the present study, bacterial growth in a rich media is analysed in a Stochastic Differential Equation (SDE) framework. It is demonstrated that the SDE formulation and smoothened state estimates provide a systematic framework for data driven model improvements, using random walk hidden states...
Large deviation tail estimates and related limit laws for stochastic fixed point equations
DEFF Research Database (Denmark)
Collamore, Jeffrey F.; Vidyashankar, Anand N.
2013-01-01
We study the forward and backward recursions generated by a stochastic fixed point equation (SFPE) of the form $V \\stackrel{d}{=} A\\max\\{V, D\\}+B$, where $(A, B, D) \\in (0, \\infty)\\times {\\mathbb R}^2$, for both the stationary and explosive cases. In the stationary case (when ${\\bf E} [\\log \\: A]...
Backward stochastic differential equations with oblique reflection and local Lipschitz drift
Directory of Open Access Journals (Sweden)
Auguste Aman
2003-01-01
Full Text Available We consider reflected backward stochastic differential equations with time and space dependent coefficients in an orthant, and with oblique reflection. Existence and uniqueness of solution are established assuming local Lipschitz continuity of the drift, Lipschitz continuity and uniform spectral radius conditions on the reflection matrix.
Fixed Points and Stability in Neutral Stochastic Differential Equations with Variable Delays
Directory of Open Access Journals (Sweden)
Chang-Wen Zhao
2008-07-01
Full Text Available We consider the mean square asymptotic stability of a generalized linear neutral stochastic differential equation with variable delays by using the fixed point theory. An asymptotic mean square stability theorem with a necessary and sufficient condition is proved, which improves and generalizes some results due to Burton, Zhang and Luo. Two examples are also given to illustrate our results.
Parametric inference for stochastic differential equations: a smooth and match approach
Gugushvili, S.; Spreij, P.
2012-01-01
We study the problem of parameter estimation for a univariate discretely observed ergodic diffusion process given as a solution to a stochastic differential equation. The estimation procedure we propose consists of two steps. In the first step, which is referred to as a smoothing step, we smooth the
A two-level stochastic collocation method for semilinear elliptic equations with random coefficients
Energy Technology Data Exchange (ETDEWEB)
Chen, Luoping; Zheng, Bin; Lin, Guang; Voulgarakis, Nikolaos
2017-05-01
In this work, we propose a novel two-level discretization for solving semilinear elliptic equations with random coefficients. Motivated by the two-grid method for deterministic partial differential equations (PDEs) introduced by Xu, our two-level stochastic collocation method utilizes a two-grid finite element discretization in the physical space and a two-level collocation method in the random domain. In particular, we solve semilinear equations on a coarse mesh $\\mathcal{T}_H$ with a low level stochastic collocation (corresponding to the polynomial space $\\mathcal{P}_{P}$) and solve linearized equations on a fine mesh $\\mathcal{T}_h$ using high level stochastic collocation (corresponding to the polynomial space $\\mathcal{P}_p$). We prove that the approximated solution obtained from this method achieves the same order of accuracy as that from solving the original semilinear problem directly by stochastic collocation method with $\\mathcal{T}_h$ and $\\mathcal{P}_p$. The two-level method is computationally more efficient, especially for nonlinear problems with high random dimensions. Numerical experiments are also provided to verify the theoretical results.
A stochastic differential equation model for drug dissolution and its parameters
Czech Academy of Sciences Publication Activity Database
Lánský, Petr; Lánská, V.; Weiss, M.
2004-01-01
Roč. 100, č. 2 (2004), s. 267-274 ISSN 0168-3659 R&D Projects: GA MZd NR7795 Grant - others:DE - CZ(CZ) CZE 01/034 Institutional research plan: CEZ:AV0Z5011922 Keywords : dissolution * stochastic differential equation Subject RIV: FR - Pharmacology ; Medidal Chemistry Impact factor: 3.297, year: 2004
DEFF Research Database (Denmark)
Debrabant, Kristian; Samaey, Giovanni; Zieliński, Przemysław
2017-01-01
We present and analyse a micro-macro acceleration method for the Monte Carlo simulation of stochastic differential equations with separation between the (fast) time-scale of individual trajectories and the (slow) time-scale of the macroscopic function of interest. The algorithm combines short...
Czech Academy of Sciences Publication Activity Database
Ondreját, Martin
2006-01-01
Roč. 6, č. 1 (2006), s. 23-52 ISSN 0219-4937 R&D Projects: GA ČR(CZ) GA201/01/1197 Institutional research plan: CEZ:AV0Z10190503 Keywords : stochastic wave equation * spatially homogeneous Wiener process * martingale solution Subject RIV: BA - General Mathematics
Directory of Open Access Journals (Sweden)
Zhanhua Yu
2011-01-01
Full Text Available We study the almost surely asymptotic stability of exact solutions to neutral stochastic pantograph equations (NSPEs, and sufficient conditions are obtained. Based on these sufficient conditions, we show that the backward Euler method (BEM with variable stepsize can preserve the almost surely asymptotic stability. Numerical examples are demonstrated for illustration.
Statistical inference for discrete-time samples from affine stochastic delay differential equations
DEFF Research Database (Denmark)
Küchler, Uwe; Sørensen, Michael
2013-01-01
Statistical inference for discrete time observations of an affine stochastic delay differential equation is considered. The main focus is on maximum pseudo-likelihood estimators, which are easy to calculate in practice. A more general class of prediction-based estimating functions is investigated...
Maddix, Danielle C.; Sampaio, Luiz; Gerritsen, Margot
2018-05-01
The degenerate parabolic Generalized Porous Medium Equation (GPME) poses numerical challenges due to self-sharpening and its sharp corner solutions. For these problems, we show results for two subclasses of the GPME with differentiable k (p) with respect to p, namely the Porous Medium Equation (PME) and the superslow diffusion equation. Spurious temporal oscillations, and nonphysical locking and lagging have been reported in the literature. These issues have been attributed to harmonic averaging of the coefficient k (p) for small p, and arithmetic averaging has been suggested as an alternative. We show that harmonic averaging is not solely responsible and that an improved discretization can mitigate these issues. Here, we investigate the causes of these numerical artifacts using modified equation analysis. The modified equation framework can be used for any type of discretization. We show results for the second order finite volume method. The observed problems with harmonic averaging can be traced to two leading error terms in its modified equation. This is also illustrated numerically through a Modified Harmonic Method (MHM) that can locally modify the critical terms to remove the aforementioned numerical artifacts.
Energy Technology Data Exchange (ETDEWEB)
Webster, Clayton G [ORNL; Zhang, Guannan [ORNL; Gunzburger, Max D [ORNL
2012-10-01
Accurate predictive simulations of complex real world applications require numerical approximations to first, oppose the curse of dimensionality and second, converge quickly in the presence of steep gradients, sharp transitions, bifurcations or finite discontinuities in high-dimensional parameter spaces. In this paper we present a novel multi-dimensional multi-resolution adaptive (MdMrA) sparse grid stochastic collocation method, that utilizes hierarchical multiscale piecewise Riesz basis functions constructed from interpolating wavelets. The basis for our non-intrusive method forms a stable multiscale splitting and thus, optimal adaptation is achieved. Error estimates and numerical examples will used to compare the efficiency of the method with several other techniques.
A Derivation of the Nonlocal Volume-Averaged Equations for Two-Phase Flow Transport
Directory of Open Access Journals (Sweden)
Gilberto Espinosa-Paredes
2012-01-01
Full Text Available In this paper a detailed derivation of the general transport equations for two-phase systems using a method based on nonlocal volume averaging is presented. The local volume averaging equations are commonly applied in nuclear reactor system for optimal design and safe operation. Unfortunately, these equations are limited to length-scale restriction and according with the theory of the averaging volume method, these fail in transition of the flow patterns and boundaries between two-phase flow and solid, which produce rapid changes in the physical properties and void fraction. The non-local volume averaging equations derived in this work contain new terms related with non-local transport effects due to accumulation, convection diffusion and transport properties for two-phase flow; for instance, they can be applied in the boundary between a two-phase flow and a solid phase, or in the boundary of the transition region of two-phase flows where the local volume averaging equations fail.
Model Identification Using Stochastic Differential Equation Grey-Box Models in Diabetes
DEFF Research Database (Denmark)
Duun-Henriksen, Anne Katrine; Schmidt, Signe; Røge, Rikke Meldgaard
2013-01-01
BACKGROUND: The acceptance of virtual preclinical testing of control algorithms is growing and thus also the need for robust and reliable models. Models based on ordinary differential equations (ODEs) can rarely be validated with standard statistical tools. Stochastic differential equations (SDEs...... are uncorrelated and provides the possibility to pinpoint model deficiencies. METHODS: An identifiable model of the glucoregulatory system in a type 1 diabetes mellitus (T1DM) patient is used as the basis for development of a stochastic-differential-equation-based grey-box model (SDE-GB). The parameters...... are estimated on clinical data from four T1DM patients. The optimal SDE-GB is determined from likelihood-ratio tests. Finally, parameter tracking is used to track the variation in the "time to peak of meal response" parameter. RESULTS: We found that the transformation of the ODE model into an SDE-GB resulted...
Vrettas, Michail D; Opper, Manfred; Cornford, Dan
2015-01-01
This work introduces a Gaussian variational mean-field approximation for inference in dynamical systems which can be modeled by ordinary stochastic differential equations. This new approach allows one to express the variational free energy as a functional of the marginal moments of the approximating Gaussian process. A restriction of the moment equations to piecewise polynomial functions, over time, dramatically reduces the complexity of approximate inference for stochastic differential equation models and makes it comparable to that of discrete time hidden Markov models. The algorithm is demonstrated on state and parameter estimation for nonlinear problems with up to 1000 dimensional state vectors and compares the results empirically with various well-known inference methodologies.
Numerical method for solving stochastic differential equations with Poissonian white shot noise.
Kim, Changho; Lee, Eok Kyun; Hänggi, Peter; Talkner, Peter
2007-07-01
We propose a numerical integration scheme to solve stochastic differential equations driven by Poissonian white shot noise. Our formula, which is based on an integral equation, which is equivalent to the stochastic differential equation, utilizes a discrete time approximation with fixed integration time step. We show that our integration formula approaches the Euler formula if the Poissonian noise approaches the Gaussian white noise. The accuracy and efficiency of the proposed algorithm are examined by studying the dynamics of an overdamped particle driven by Poissonian white shot noise in a spatially periodic potential. We find that the accuracy of the proposed algorithm only weakly depends on the parameters characterizing the Poissonian white shot noise; this holds true even if the limit of Gaussian white noise is approached.
Energy Technology Data Exchange (ETDEWEB)
Lee, Kok Foong [Department of Chemical Engineering and Biotechnology, University of Cambridge, New Museums Site, Pembroke Street, Cambridge CB2 3RA (United Kingdom); Patterson, Robert I.A.; Wagner, Wolfgang [Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstraße 39, 10117 Berlin (Germany); Kraft, Markus, E-mail: mk306@cam.ac.uk [Department of Chemical Engineering and Biotechnology, University of Cambridge, New Museums Site, Pembroke Street, Cambridge CB2 3RA (United Kingdom); School of Chemical and Biomedical Engineering, Nanyang Technological University, 62 Nanyang Drive, Singapore, 637459 (Singapore)
2015-12-15
Graphical abstract: -- Highlights: •Problems concerning multi-compartment population balance equations are studied. •A class of fragmentation weight transfer functions is presented. •Three stochastic weighted algorithms are compared against the direct simulation algorithm. •The numerical errors of the stochastic solutions are assessed as a function of fragmentation rate. •The algorithms are applied to a multi-dimensional granulation model. -- Abstract: This paper introduces stochastic weighted particle algorithms for the solution of multi-compartment population balance equations. In particular, it presents a class of fragmentation weight transfer functions which are constructed such that the number of computational particles stays constant during fragmentation events. The weight transfer functions are constructed based on systems of weighted computational particles and each of it leads to a stochastic particle algorithm for the numerical treatment of population balance equations. Besides fragmentation, the algorithms also consider physical processes such as coagulation and the exchange of mass with the surroundings. The numerical properties of the algorithms are compared to the direct simulation algorithm and an existing method for the fragmentation of weighted particles. It is found that the new algorithms show better numerical performance over the two existing methods especially for systems with significant amount of large particles and high fragmentation rates.
Wood, Julie; Oravecz, Zita; Vogel, Nina; Benson, Lizbeth; Chow, Sy-Miin; Cole, Pamela; Conroy, David E; Pincus, Aaron L; Ram, Nilam
2017-12-15
Life-span theories of aging suggest improvements and decrements in individuals' ability to regulate affect. Dynamic process models, with intensive longitudinal data, provide new opportunities to articulate specific theories about individual differences in intraindividual dynamics. This paper illustrates a method for operationalizing affect dynamics using a multilevel stochastic differential equation (SDE) model, and examines how those dynamics differ with age and trait-level tendencies to deploy emotion regulation strategies (reappraisal and suppression). Univariate multilevel SDE models, estimated in a Bayesian framework, were fit to 21 days of ecological momentary assessments of affect valence and arousal (average 6.93/day, SD = 1.89) obtained from 150 adults (age 18-89 years)-specifically capturing temporal dynamics of individuals' core affect in terms of attractor point, reactivity to biopsychosocial (BPS) inputs, and attractor strength. Older age was associated with higher arousal attractor point and less BPS-related reactivity. Greater use of reappraisal was associated with lower valence attractor point. Intraindividual variability in regulation strategy use was associated with greater BPS-related reactivity and attractor strength, but in different ways for valence and arousal. The results highlight the utility of SDE models for studying affect dynamics and informing theoretical predictions about how intraindividual dynamics change over the life course. © The Author 2017. Published by Oxford University Press on behalf of The Gerontological Society of America. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com.
Fractional diffusion equation with distributed-order material derivative. Stochastic foundations
International Nuclear Information System (INIS)
Magdziarz, M; Teuerle, M
2017-01-01
In this paper, we present the stochastic foundations of fractional dynamics driven by the fractional material derivative of distributed-order type. Before stating our main result, we present the stochastic scenario which underlies the dynamics given by the fractional material derivative. Then we introduce the Lévy walk process of distributed-order type to establish our main result, which is the scaling limit of the considered process. It appears that the probability density function of the scaling limit process fulfills, in a weak sense, the fractional diffusion equation with the material derivative of distributed-order type. (paper)
Mean-Variance Hedging and Forward-Backward Stochastic Differential Filtering Equations
Directory of Open Access Journals (Sweden)
Guangchen Wang
2011-01-01
Full Text Available This paper is concerned with a mean-variance hedging problem with partial information, where the initial endowment of an agent may be a decision and the contingent claim is a random variable. This problem is explicitly solved by studying a linear-quadratic optimal control problem with non-Markov control systems and partial information. Then, we use the result as well as filtering to solve some examples in stochastic control and finance. Also, we establish backward and forward-backward stochastic differential filtering equations which are different from the classical filtering theory introduced by Liptser and Shiryayev (1977, Xiong (2008, and so forth.
Probabilistic Forecast of Wind Power Generation by Stochastic Differential Equation Models
Elkantassi, Soumaya
2017-04-01
Reliable forecasting of wind power generation is crucial to optimal control of costs in generation of electricity with respect to the electricity demand. Here, we propose and analyze stochastic wind power forecast models described by parametrized stochastic differential equations, which introduce appropriate fluctuations in numerical forecast outputs. We use an approximate maximum likelihood method to infer the model parameters taking into account the time correlated sets of data. Furthermore, we study the validity and sensitivity of the parameters for each model. We applied our models to Uruguayan wind power production as determined by historical data and corresponding numerical forecasts for the period of March 1 to May 31, 2016.
Hutzenthaler, Martin
2015-01-01
Many stochastic differential equations (SDEs) in the literature have a superlinearly growing nonlinearity in their drift or diffusion coefficient. Unfortunately, moments of the computationally efficient Euler-Maruyama approximation method diverge for these SDEs in finite time. This article develops a general theory based on rare events for studying integrability properties such as moment bounds for discrete-time stochastic processes. Using this approach, the authors establish moment bounds for fully and partially drift-implicit Euler methods and for a class of new explicit approximation method
Stochastic solutions of Navier-Stokes equations: an experimental evidence.
Djurek, Ivan; Djurek, Danijel; Petosić, Antonio
2010-12-01
An electrodynamic loudspeaker has been operated in anharmonic regime indicated by the nonlinear ordinary differential equation when spring constant γ in restoring term, as well as, viscoelasticity of the membrane material, increases with displacement. For driving currents in the range of 2.8-3.3 A, doubling of the vibration period appears, while for currents in the range of 3.3-3.6 A, multiple sequences of subharmonic vibrations begin with f/4 and 3f/4. An application of currents higher than 3.6 A results in a spectrum, characteristic for the chaotic state. The loudspeaker was then operated in a closed chamber, and subharmonic vibrations disappeared by an evacuation. Subsequent injection of air revoked them again at ∼ 120 mbar (Re(')=476) when air viscous forces dominate the Morse convection. At 430 mbar (Re=538) single vibration state was restored, and the phenomenon is in an agreement with prediction of the five mode truncation procedure applied to the Navier-Stokes equations describing a two-dimensional incompressible fluid. © 2010 American Institute of Physics.
Modeling bacterial clearance using stochastic-differential equations.
Atalla, Ashraf; Jeremic, Aleksandar
2010-01-01
Capillary - tissue fluid exchange is controlled by the blood pressure in the capillary and the osmotic pressure of blood (pressure of the tissue fluid outside the capillaries). In this paper, we develop a mathematical model to simulate the movement of bacteria into and within a capillary segment. The model is based on Fokker-Planck equation and Navier-Stocks equations that accounts for different boundary conditions. Also, we model the transportation through capillary walls by means of anisotropic diffusivity that depends on the pressure difference across the capillary walls. By solving the model with a numerical method, it was possible to predict the concentration of bacteria at points within the capillary. However, numerical analysis consumes computational time and resources. To efficiently simulate the bacterial clearance, we propose a segmentation model that is based on breaking the capillary network into smaller sections with pre-defined properties in order to reduce the overall computational time. The proposed model shows a great reduction in computational time and provides accurate results when compared to the numerical analysis.
Yosida approximations of stochastic differential equations in infinite dimensions and applications
Govindan, T E
2016-01-01
This research monograph brings together, for the first time, the varied literature on Yosida approximations of stochastic differential equations (SDEs) in infinite dimensions and their applications into a single cohesive work. The author provides a clear and systematic introduction to the Yosida approximation method and justifies its power by presenting its applications in some practical topics such as stochastic stability and stochastic optimal control. The theory assimilated spans more than 35 years of mathematics, but is developed slowly and methodically in digestible pieces. The book begins with a motivational chapter that introduces the reader to several different models that play recurring roles throughout the book as the theory is unfolded, and invites readers from different disciplines to see immediately that the effort required to work through the theory that follows is worthwhile. From there, the author presents the necessary prerequisite material, and then launches the reader into the main discussi...
Tamellini, L.
2014-01-01
In this paper we consider a proper generalized decomposition method to solve the steady incompressible Navier-Stokes equations with random Reynolds number and forcing term. The aim of such a technique is to compute a low-cost reduced basis approximation of the full stochastic Galerkin solution of the problem at hand. A particular algorithm, inspired by the Arnoldi method for solving eigenproblems, is proposed for an efficient greedy construction of a deterministic reduced basis approximation. This algorithm decouples the computation of the deterministic and stochastic components of the solution, thus allowing reuse of preexisting deterministic Navier-Stokes solvers. It has the remarkable property of only requiring the solution of m uncoupled deterministic problems for the construction of an m-dimensional reduced basis rather than M coupled problems of the full stochastic Galerkin approximation space, with m l M (up to one order of magnitudefor the problem at hand in this work). © 2014 Society for Industrial and Applied Mathematics.
Jimenez, M. Navarro
2017-04-18
A Galerkin polynomial chaos (PC) method was recently proposed to perform variance decomposition and sensitivity analysis in stochastic differential equations (SDEs), driven by Wiener noise and involving uncertain parameters. The present paper extends the PC method to nonintrusive approaches enabling its application to more complex systems hardly amenable to stochastic Galerkin projection methods. We also discuss parallel implementations and the variance decomposition of the derived quantity of interest within the framework of nonintrusive approaches. In particular, a novel hybrid PC-sampling-based strategy is proposed in the case of nonsmooth quantities of interest (QoIs) but smooth SDE solution. Numerical examples are provided that illustrate the decomposition of the variance of QoIs into contributions arising from the uncertain parameters, the inherent stochastic forcing, and joint effects. The simulations are also used to support a brief analysis of the computational complexity of the method, providing insight on the types of problems that would benefit from the present developments.
Khan, Sami Ullah; Ali, Ishtiaq
2018-03-01
Explicit solutions to delay differential equation (DDE) and stochastic delay differential equation (SDDE) can rarely be obtained, therefore numerical methods are adopted to solve these DDE and SDDE. While on the other hand due to unstable nature of both DDE and SDDE numerical solutions are also not straight forward and required more attention. In this study, we derive an efficient numerical scheme for DDE and SDDE based on Legendre spectral-collocation method, which proved to be numerical methods that can significantly speed up the computation. The method transforms the given differential equation into a matrix equation by means of Legendre collocation points which correspond to a system of algebraic equations with unknown Legendre coefficients. The efficiency of the proposed method is confirmed by some numerical examples. We found that our numerical technique has a very good agreement with other methods with less computational effort.
Directory of Open Access Journals (Sweden)
Jha Sumit
2012-04-01
Full Text Available Abstract Stochastic Differential Equations (SDE are often used to model the stochastic dynamics of biological systems. Unfortunately, rare but biologically interesting behaviors (e.g., oncogenesis can be difficult to observe in stochastic models. Consequently, the analysis of behaviors of SDE models using numerical simulations can be challenging. We introduce a method for solving the following problem: given a SDE model and a high-level behavioral specification about the dynamics of the model, algorithmically decide whether the model satisfies the specification. While there are a number of techniques for addressing this problem for discrete-state stochastic models, the analysis of SDE and other continuous-state models has received less attention. Our proposed solution uses a combination of Bayesian sequential hypothesis testing, non-identically distributed samples, and Girsanov's theorem for change of measures to examine rare behaviors. We use our algorithm to analyze two SDE models of tumor dynamics. Our use of non-identically distributed samples sampling contributes to the state of the art in statistical verification and model checking of stochastic models by providing an effective means for exposing rare events in SDEs, while retaining the ability to compute bounds on the probability that those events occur.
Jha, Sumit Kumar; Langmead, Christopher James
2012-04-12
Stochastic Differential Equations (SDE) are often used to model the stochastic dynamics of biological systems. Unfortunately, rare but biologically interesting behaviors (e.g., oncogenesis) can be difficult to observe in stochastic models. Consequently, the analysis of behaviors of SDE models using numerical simulations can be challenging. We introduce a method for solving the following problem: given a SDE model and a high-level behavioral specification about the dynamics of the model, algorithmically decide whether the model satisfies the specification. While there are a number of techniques for addressing this problem for discrete-state stochastic models, the analysis of SDE and other continuous-state models has received less attention. Our proposed solution uses a combination of Bayesian sequential hypothesis testing, non-identically distributed samples, and Girsanov's theorem for change of measures to examine rare behaviors. We use our algorithm to analyze two SDE models of tumor dynamics. Our use of non-identically distributed samples sampling contributes to the state of the art in statistical verification and model checking of stochastic models by providing an effective means for exposing rare events in SDEs, while retaining the ability to compute bounds on the probability that those events occur.
Exact solutions for the Wick-type stochastic Kersten-Krasil'shchik coupled KdV-mKdV equations
Singh, S.; Saha Ray, S.
2017-11-01
In this article, exact solutions of Wick-type stochastic Kersten-Krasil'shchik coupled KdV-mKdV equations have been obtained by using the Jacobian elliptic function expansion method. We have used the Hermite transform for transforming the Wick-type stochastic Kersten-Krasil'shchik coupled KdV-mKdV equation into a deterministic partial differential equation. Also, we have applied the inverse Hermite transform for obtaining a set of stochastic solutions in the white noise space.
Directory of Open Access Journals (Sweden)
Xiao-Li Ding
2018-01-01
Full Text Available In this paper, we investigate analytical solutions of multi-time scale fractional stochastic differential equations driven by fractional Brownian motions. We firstly decompose homogeneous multi-time scale fractional stochastic differential equations driven by fractional Brownian motions into independent differential subequations, and give their analytical solutions. Then, we use the variation of constant parameters to obtain the solutions of nonhomogeneous multi-time scale fractional stochastic differential equations driven by fractional Brownian motions. Finally, we give three examples to demonstrate the applicability of our obtained results.
Conditional stability in determination of initial data for stochastic parabolic equations
International Nuclear Information System (INIS)
Yuan, Ganghua
2017-01-01
In this paper, we solve two kinds of inverse problems in determination of the initial data for stochastic parabolic equations. One is determination of the initial data by lateral boundary observation on arbitrary portion of the boundary, the second one is determination of the initial data by internal observation in a subregion inside the domain. We obtain conditional stability for the two kinds of inverse problems. To prove the results, we estimate the initial data by a terminal observation near the initial time, then we estimate this terminal observation by lateral boundary observation on arbitrary portion of the boundary or internal observation in a subregion inside the domain. To achieve those goals, we derive several new Carleman estimates for stochastic parabolic equations in this paper. (paper)
Conditional stability in determination of initial data for stochastic parabolic equations
Yuan, Ganghua
2017-03-01
In this paper, we solve two kinds of inverse problems in determination of the initial data for stochastic parabolic equations. One is determination of the initial data by lateral boundary observation on arbitrary portion of the boundary, the second one is determination of the initial data by internal observation in a subregion inside the domain. We obtain conditional stability for the two kinds of inverse problems. To prove the results, we estimate the initial data by a terminal observation near the initial time, then we estimate this terminal observation by lateral boundary observation on arbitrary portion of the boundary or internal observation in a subregion inside the domain. To achieve those goals, we derive several new Carleman estimates for stochastic parabolic equations in this paper.
Cellular ion channel-pump system modeling using switched stochastic differential equations.
Weaver, Jeffrey
2007-01-01
This paper identifies a multidimensional random switched process model of a neuron with embedded Ca++ ion channel and pump molecules adiabatically interacting based on local ion concentrations near the cell membrane. The model interprets known physiology of the channels as a coupled set of switched random processes and derives mechanical equations based on concentration flow among different states of the system. Rapid changes to channel barrier energies occurring during channel opening and closing transitions are modeled as another degree of freedom commutating the state of the overall system. An ion reservoir model is used as the primary tool to incorporate stochastic effects in channel operation. The complete model is analyzed numerically and then the equations are used to motivate a stochastic model for closed state dwell times. The result is compared against expected results of a leaky-integrator and known single-channel histograms.
Functional Integral Approach to the Solution of a System of Stochastic Differential Equations
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Ayryan Edik
2018-01-01
Full Text Available A new method for the evaluation of the characteristics of the solution of a system of stochastic differential equations is presented. This method is based on the representation of a probability density function p through a functional integral. The functional integral representation is obtained by means of the Onsager-Machlup functional technique for a special case when the diffusion matrix for the SDE system defines a Riemannian space with zero curvature.
Cu(II)-porphyrin molecular dynamics as seen in novel EPR/stochastic Liouville equation study
Hakansson, Par; Nguyen, ThaoNguyen; Nair, Prasanth B; Edge, Ruth; Stulz, Eugen
2013-01-01
Copper porphyrin dissolved in CH2Cl2:Toluene as fluid and frozen solution was studied as a function of temperature with X-band electron paramagnetic resonance (EPR). Quantitative interpretation was obtained with a recently developed Stochastic Liouville simulation method. For the first time we address the large spin system that translates into a 400 000 dimensional Liouville equation solved under slow-motion conditions. With a simple three parameter microscopic model, the temperature dependen...
Directory of Open Access Journals (Sweden)
Xiaolin Zhu
2014-01-01
Full Text Available This paper studies the T-stability of the Heun method and balanced method for solving stochastic differential delay equations (SDDEs. Two T-stable conditions of the Heun method are obtained for two kinds of linear SDDEs. Moreover, two conditions under which the balanced method is T-stable are obtained for two kinds of linear SDDEs. Some numerical examples verify the theoretical results proposed.
Weak solutions to stochastic differential equations driven by fractional Brownian motion
Czech Academy of Sciences Publication Activity Database
Šnupárková, Jana
2009-01-01
Roč. 59, č. 4 (2009), s. 879-907 ISSN 0011-4642 R&D Projects: GA ČR GA201/07/0237 Institutional research plan: CEZ:AV0Z10750506 Keywords : fractional Brownian motion * weak solutions Subject RIV: BA - General Mathematics Impact factor: 0.306, year: 2009 http://library.utia.cas.cz/separaty/2009/SI/snuparkova-weak solutions to stochastic differential equations driven by fractional brownian motion.pdf
Stochastic Differential Equations and Markov Processes in the Modeling of Electrical Circuits
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R. Rezaeyan
2010-06-01
Full Text Available Stochastic differential equations(SDEs, arise from physical systems that possess inherent noise and certainty. We derive a SDE for electrical circuits. In this paper, we will explore the close relationship between the SDE and autoregressive(AR model. We will solve SDE related to RC circuit with using of AR(1 model (Markov process and however with Euler-Maruyama(EM method. Then, we will compare this solutions. Numerical simulations in MATLAB are obtained.
Functional Integral Approach to the Solution of a System of Stochastic Differential Equations
Ayryan, Edik; Egorov, Alexander; Kulyabov, Dmitri; Malyutin, Victor; Sevastianov, Leonid
2018-02-01
A new method for the evaluation of the characteristics of the solution of a system of stochastic differential equations is presented. This method is based on the representation of a probability density function p through a functional integral. The functional integral representation is obtained by means of the Onsager-Machlup functional technique for a special case when the diffusion matrix for the SDE system defines a Riemannian space with zero curvature.
Invariant measure for the stochastic Navier-Stokes equations in unbounded 2D domains
Czech Academy of Sciences Publication Activity Database
Brzezniak, Z.; Motyl, E.; Ondreját, Martin
2017-01-01
Roč. 45, č. 5 (2017), s. 3145-3201 ISSN 0091-1798 R&D Projects: GA ČR(CZ) GA15-08819S Institutional support: RVO:67985556 Keywords : Invariant measure * bw-Feller semigroup * stochastic Navier–Stokes equation Subject RIV: BA - General Mathematics OBOR OECD: Statistics and probability Impact factor: 1.940, year: 2016 http://library.utia.cas.cz/separaty/2017/SI/ondrejat-0478383.pdf
Mélykúti, Bence
2010-01-01
The Chemical Langevin Equation (CLE), which is a stochastic differential equation driven by a multidimensional Wiener process, acts as a bridge between the discrete stochastic simulation algorithm and the deterministic reaction rate equation when simulating (bio)chemical kinetics. The CLE model is valid in the regime where molecular populations are abundant enough to assume their concentrations change continuously, but stochastic fluctuations still play a major role. The contribution of this work is that we observe and explore that the CLE is not a single equation, but a parametric family of equations, all of which give the same finite-dimensional distribution of the variables. On the theoretical side, we prove that as many Wiener processes are sufficient to formulate the CLE as there are independent variables in the equation, which is just the rank of the stoichiometric matrix. On the practical side, we show that in the case where there are m1 pairs of reversible reactions and m2 irreversible reactions there is another, simple formulation of the CLE with only m1 + m2 Wiener processes, whereas the standard approach uses 2 m1 + m2. We demonstrate that there are considerable computational savings when using this latter formulation. Such transformations of the CLE do not cause a loss of accuracy and are therefore distinct from model reduction techniques. We illustrate our findings by considering alternative formulations of the CLE for a human ether a-go-go related gene ion channel model and the Goldbeter-Koshland switch. © 2010 American Institute of Physics.
Rackauckas, Christopher; Nie, Qing
2017-01-01
Adaptive time-stepping with high-order embedded Runge-Kutta pairs and rejection sampling provides efficient approaches for solving differential equations. While many such methods exist for solving deterministic systems, little progress has been made for stochastic variants. One challenge in developing adaptive methods for stochastic differential equations (SDEs) is the construction of embedded schemes with direct error estimates. We present a new class of embedded stochastic Runge-Kutta (SRK) methods with strong order 1.5 which have a natural embedding of strong order 1.0 methods. This allows for the derivation of an error estimate which requires no additional function evaluations. Next we derive a general method to reject the time steps without losing information about the future Brownian path termed Rejection Sampling with Memory (RSwM). This method utilizes a stack data structure to do rejection sampling, costing only a few floating point calculations. We show numerically that the methods generate statistically-correct and tolerance-controlled solutions. Lastly, we show that this form of adaptivity can be applied to systems of equations, and demonstrate that it solves a stiff biological model 12.28x faster than common fixed timestep algorithms. Our approach only requires the solution to a bridging problem and thus lends itself to natural generalizations beyond SDEs.
Shannon, Robin; Glowacki, David R
2018-02-15
The chemical master equation is a powerful theoretical tool for analyzing the kinetics of complex multiwell potential energy surfaces in a wide range of different domains of chemical kinetics spanning combustion, atmospheric chemistry, gas-surface chemistry, solution phase chemistry, and biochemistry. There are two well-established methodologies for solving the chemical master equation: a stochastic "kinetic Monte Carlo" approach and a matrix-based approach. In principle, the results yielded by both approaches are identical; the decision of which approach is better suited to a particular study depends on the details of the specific system under investigation. In this Article, we present a rigorous method for accelerating stochastic approaches by several orders of magnitude, along with a method for unbiasing the accelerated results to recover the "true" value. The approach we take in this paper is inspired by the so-called "boxed molecular dynamics" (BXD) method, which has previously only been applied to accelerate rare events in molecular dynamics simulations. Here we extend BXD to design a simple algorithmic strategy for accelerating rare events in stochastic kinetic simulations. Tests on a number of systems show that the results obtained using the BXD rare event strategy are in good agreement with unbiased results. To carry out these tests, we have implemented a kinetic Monte Carlo approach in MESMER, which is a cross-platform, open-source, and freely available master equation solver.
Numerical solution of the stochastic parabolic equation with the dependent operator coefficient
Energy Technology Data Exchange (ETDEWEB)
Ashyralyev, Allaberen [Department of Elementary Mathematics Education, Fatih University, 34500, Istanbul (Turkey); Department of Mathematics, ITTU, Ashgabat (Turkmenistan); Okur, Ulker [Institute for Stochastics and Applications, Department of Mathematics, University of Stuttgart, 70569, Stuttgart (Germany)
2015-09-18
In the present paper, a single step implicit difference scheme for the numerical solution of the stochastic parabolic equation with the dependent operator coefficient is presented. Theorem on convergence estimates for the solution of this difference scheme is established. In applications, this abstract result permits us to obtain the convergence estimates for the solution of difference schemes for the numerical solution of initial boundary value problems for parabolic equations. The theoretical statements for the solution of this difference scheme are supported by the results of numerical experiments.
Pfeffer, Tobias; Pollet, Lode
2017-04-01
We present the construction and stochastic summation of rooted-tree diagrams, based on the expansion of a root finding algorithm applied to the Dyson-Schwinger equations. The mathematical formulation shows superior convergence properties compared to the bold diagrammatic Monte Carlo approach and the developed algorithm allows one to tackle generic high-dimensional integral equations, to avoid the curse of dealing explicitly with high-dimensional objects and to access non-perturbative regimes. The sign problem remains the limiting factor, but it is not found to be worse than in other approaches. We illustrate the method for {φ }4 theory but note that it applies in principle to any model.
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Bixiang Wang
2013-08-01
Full Text Available We prove the existence and uniqueness of random attractors for the p-Laplace equation driven simultaneously by non-autonomous deterministic and stochastic forcing. The nonlinearity of the equation is allowed to have a polynomial growth rate of any order which may be greater than p. We further establish the upper semicontinuity of random attractors as the intensity of noise approaches zero. In addition, we show the pathwise periodicity of random attractors when all non-autonomous deterministic forcing terms are time periodic.
On stochastic differential equations driven by the renormalized square of the Gaussian white noise
Ben Ammou, Bilel Kacem; Lanconelli, Alberto
2015-11-01
We investigate the properties of the Wick square of Gaussian white noises through a new method to perform nonlinear operations on Hida distributions. This method lays in between the Wick product interpretation and the usual definition of nonlinear functions. We prove an Itô-type formula and solve stochastic differential equations driven by the renormalized square of the Gaussian white noise. Our approach works with standard assumptions on the coefficients of the equations, global Lipschitz continuity, and produces existence and uniqueness results in the space where the noise lives. The linear case is studied in details and positivity of the solution is proved.
Bound states of quarks calculated with stochastic integration of the Bethe-Salpeter equation
International Nuclear Information System (INIS)
Salomon, M.
1992-07-01
We have computed the masses, wave functions and sea quark content of mesons in their ground state by integrating the Bethe-Salpeter equation with a stochastic algorithm. This method allows the inclusion of a large set of diagrams. Inspection of the kernel of the equation shows that q-q-bar pairs with similar constituent masses in a singlet spin state exhibit a high bound state which is not present in other pairs. The pion, kaon and eta belongs to this category. 19 refs., 2 figs., 2 tabs
DEFF Research Database (Denmark)
Møller, Jan Kloppenborg; Madsen, Henrik
This report describes methods to eliminate state dependent diffusion terms in Stochastic Differential Equations (SDEs). Transformations that leave the diffusion term of SDEs constant is important for simulation, and estimation. It is important for simulation because the Euler approximation...... convergence rate is faster, and for estimation because the Extended Kalman Filter equations are easier to implement than higher order filters needed in the case of state dependent diffusion terms. The general class of transformations which leaves the diffusion term independent of the state is called...
International Nuclear Information System (INIS)
Brett, Tobias; Galla, Tobias
2014-01-01
We present a heuristic derivation of Gaussian approximations for stochastic chemical reaction systems with distributed delay. In particular, we derive the corresponding chemical Langevin equation. Due to the non-Markovian character of the underlying dynamics, these equations are integro-differential equations, and the noise in the Gaussian approximation is coloured. Following on from the chemical Langevin equation, a further reduction leads to the linear-noise approximation. We apply the formalism to a delay variant of the celebrated Brusselator model, and show how it can be used to characterise noise-driven quasi-cycles, as well as noise-triggered spiking. We find surprisingly intricate dependence of the typical frequency of quasi-cycles on the delay period
Brett, Tobias; Galla, Tobias
2014-03-28
We present a heuristic derivation of Gaussian approximations for stochastic chemical reaction systems with distributed delay. In particular, we derive the corresponding chemical Langevin equation. Due to the non-Markovian character of the underlying dynamics, these equations are integro-differential equations, and the noise in the Gaussian approximation is coloured. Following on from the chemical Langevin equation, a further reduction leads to the linear-noise approximation. We apply the formalism to a delay variant of the celebrated Brusselator model, and show how it can be used to characterise noise-driven quasi-cycles, as well as noise-triggered spiking. We find surprisingly intricate dependence of the typical frequency of quasi-cycles on the delay period.
DEFF Research Database (Denmark)
Ditlevsen, Susanne; Samson, Adeline
2016-01-01
Dynamics of the membrane potential in a single neuron can be studied by estimating biophysical parameters from intracellular recordings. Diffusion processes, given as continuous solutions to stochastic differential equations, are widely applied as models for the neuronal membrane potential evolut...
A doubly averaging method for third body perturbations in planet equator coordinates
Hwok, Johnny H.
1992-01-01
The first order doubly averaged potential due to third-body gravity is derived in any arbitrary coordinates. The equations of motion are nonsingular at zero eccentricity. The derivation uses a recursive method which allows easy expansion to higher order terms. Instead of using analytical quadrature to obtain the doubly averaged potential, the method presented in this paper simply eliminates the mean anomaly of the perturbed and perturbing bodies by inspection of the recursive formulation. The derivatives of the orbital elements can be numerically integrated rapidly. When a planet equator coordinate system is used, they can be added directly to the derivatives due to gravity harmonics without any coordinate transformation. The method is applied to various high altitude missions. The results are compared with a high precision numerical integration method and are found to provide excellent agreement.
Stochastic models with power-law tails the equation X = AX + B
Buraczewski, Dariusz; Mikosch, Thomas
2016-01-01
In this monograph the authors give a systematic approach to the probabilistic properties of the fixed point equation X=AX+B. A probabilistic study of the stochastic recurrence equation X_t=A_tX_{t-1}+B_t for real- and matrix-valued random variables A_t, where (A_t,B_t) constitute an iid sequence, is provided. The classical theory for these equations, including the existence and uniqueness of a stationary solution, the tail behavior with special emphasis on power law behavior, moments and support, is presented. The authors collect recent asymptotic results on extremes, point processes, partial sums (central limit theory with special emphasis on infinite variance stable limit theory), large deviations, in the univariate and multivariate cases, and they further touch on the related topics of smoothing transforms, regularly varying sequences and random iterative systems. The text gives an introduction to the Kesten-Goldie theory for stochastic recurrence equations of the type X_t=A_tX_{t-1}+B_t. It provides the c...
International Nuclear Information System (INIS)
Harko, Tiberiu; Leung, Chun Sing; Mocanu, Gabriela
2014-01-01
We consider a description of the stochastic oscillations of the general relativistic accretion disks around compact astrophysical objects interacting with their external medium based on a generalized Langevin equation with colored noise and on the fluctuation-dissipation theorems. The former accounts for the general memory and retarded effects of the frictional force. The presence of the memory effects influences the response of the disk to external random interactions, and it modifies the dynamical behavior of the disk, as well as the energy dissipation processes. The generalized Langevin equation of the motion of the disk in the vertical direction is studied numerically, and the vertical displacements, velocities, and luminosities of the stochastically perturbed disks are explicitly obtained for both the Schwarzschild and the Kerr cases. The power spectral distribution of the disk luminosity is also obtained. As a possible astrophysical application of the formalism we investigate the possibility that the intra-day variability of the active galactic nuclei may be due to the stochastic disk instabilities. The perturbations due to colored/nontrivially correlated noise induce a complicated disk dynamics, which could explain some astrophysical observational features related to disk variability. (orig.)
Harko, Tiberiu; Leung, Chun Sing; Mocanu, Gabriela
2014-05-01
We consider a description of the stochastic oscillations of the general relativistic accretion disks around compact astrophysical objects interacting with their external medium based on a generalized Langevin equation with colored noise and on the fluctuation-dissipation theorems. The former accounts for the general memory and retarded effects of the frictional force. The presence of the memory effects influences the response of the disk to external random interactions, and it modifies the dynamical behavior of the disk, as well as the energy dissipation processes. The generalized Langevin equation of the motion of the disk in the vertical direction is studied numerically, and the vertical displacements, velocities, and luminosities of the stochastically perturbed disks are explicitly obtained for both the Schwarzschild and the Kerr cases. The power spectral distribution of the disk luminosity is also obtained. As a possible astrophysical application of the formalism we investigate the possibility that the intra-day variability of the active galactic nuclei may be due to the stochastic disk instabilities. The perturbations due to colored/nontrivially correlated noise induce a complicated disk dynamics, which could explain some astrophysical observational features related to disk variability.
Numerical Resolution of N-dimensional Fokker-Planck stochastic equations
International Nuclear Information System (INIS)
Garcia-Olivares, R. A.; Munoz Roldan, A.
1992-01-01
This document describes the use of a library of programs able to solve stochastic Fokker-Planck equations in a N-dimensional space. The input data are essentially: (i) the initial distribution of the stochastic variable, (ii) the drift and fluctuation coefficients as a function of the state (which can be obtained from the transition probabilities between neighboring states) and (iii) some parameters controlling the run. A last version of the library accepts sources and sinks defined in the states space. The output is the temporal evolution of the probability distribution in the space defined by a N-dimensional grid. Some applications and readings in Synergetic, Self-Organization, transport phenomena, Ecology and other fields are suggested. If the probability distribution is interpreted as a distribution of particles then the codes can be used to solve the N-dimensional problem of advection-diffusion. (Author) 16 refs
A Posteriori Error Analysis of Stochastic Differential Equations Using Polynomial Chaos Expansions
Butler, T.
2011-01-01
We develop computable a posteriori error estimates for linear functionals of a solution to a general nonlinear stochastic differential equation with random model/source parameters. These error estimates are based on a variational analysis applied to stochastic Galerkin methods for forward and adjoint problems. The result is a representation for the error estimate as a polynomial in the random model/source parameter. The advantage of this method is that we use polynomial chaos representations for the forward and adjoint systems to cheaply produce error estimates by simple evaluation of a polynomial. By comparison, the typical method of producing such estimates requires repeated forward/adjoint solves for each new choice of random parameter. We present numerical examples showing that there is excellent agreement between these methods. © 2011 Society for Industrial and Applied Mathematics.
Numerical resolution of N-dimensional Fokker-Plank stochastic equations
International Nuclear Information System (INIS)
Garcia-Olivares, A.; Muoz, A.
1992-01-01
This document describes the use of a library of programs able to solve stochastic Fokker-Planck equations in a N-dimensional space. the input data are essentially: (i) the initial distribution of the stochastic variable, (ii) the drift and fluctuation coefficients as a function of the state (which can be obtained from the transition probabilities between neighboring states) and (iii) some parameters controlling the run. A last version of the library accepts sources and sinks defined in the states space. The output is the temporal evolution of the probability distribution in the space defined by a N-dimensional grid. Some applications and readings in Synergetics, Self-Organization, transport phenomena, Ecology and other fields are suggested. If the probability distribution is interpreted as a distribution of particles then the codes can be used to solve the N-dimensional problem of advection-diffusion. (author) 21 fig. 16 ref
Higher-order Solution of Stochastic Diffusion equation with Nonlinear Losses Using WHEP technique
El-Beltagy, Mohamed A.
2014-01-06
Using Wiener-Hermite expansion with perturbation (WHEP) technique in the solution of the stochastic partial differential equations (SPDEs) has the advantage of converting the problem to a system of deterministic equations that can be solved efficiently using the standard deterministic numerical methods [1]. The Wiener-Hermite expansion is the only known expansion that handles the white/colored noise exactly. The main statistics, such as the mean, covariance, and higher order statistical moments, can be calculated by simple formulae involving only the deterministic Wiener-Hermite coefficients. In this poster, the WHEP technique is used to solve the 2D diffusion equation with nonlinear losses and excited with white noise. The solution will be obtained numerically and will be validated and compared with the analytical solution that can be obtained from any symbolic mathematics package such as Mathematica.
Zhou, Shengfan
2017-11-01
In this paper, we first improve the existing conditions for the existence of a random exponential attractor for a continuous cocycle on a separable Banach space. Then we consider the existence of a random exponential attractor for stochastic non-autonomous reaction-diffusion equation with multiplicative noise defined in R3, which implies the existence of a random attractor with finite fractal dimension. The essential difficulty here is the continuity of the spectrum of the linear part of the equation, which can be overcome by the "tail" estimation of solutions of equation and carefully decomposing the solution into a sum of three parts, of whose, one part is finite-dimensional and other two parts are "quickly decay" in mean sense.
Stochastic differential equation models for ion channel noise in Hodgkin-Huxley neurons
Goldwyn, Joshua H.; Imennov, Nikita S.; Famulare, Michael; Shea-Brown, Eric
2011-04-01
The random transitions of ion channels between conducting and nonconducting states generate a source of internal fluctuations in a neuron, known as channel noise. The standard method for modeling the states of ion channels nonlinearly couples continuous-time Markov chains to a differential equation for voltage. Beginning with the work of R. F. Fox and Y.-N. Lu [Phys. Rev. EMTHDE91539-375510.1103/PhysRevE.49.3421 49, 3421 (1994)], there have been attempts to generate simpler models that use stochastic differential equation (SDEs) to approximate the stochastic spiking activity produced by Markov chain models. Recent numerical investigations, however, have raised doubts that SDE models can capture the stochastic dynamics of Markov chain models.We analyze three SDE models that have been proposed as approximations to the Markov chain model: one that describes the states of the ion channels and two that describe the states of the ion channel subunits. We show that the former channel-based approach can capture the distribution of channel noise and its effects on spiking in a Hodgkin-Huxley neuron model to a degree not previously demonstrated, but the latter two subunit-based approaches cannot. Our analysis provides intuitive and mathematical explanations for why this is the case. The temporal correlation in the channel noise is determined by the combinatorics of bundling subunits into channels, but the subunit-based approaches do not correctly account for this structure. Our study confirms and elucidates the findings of previous numerical investigations of subunit-based SDE models. Moreover, it presents evidence that Markov chain models of the nonlinear, stochastic dynamics of neural membranes can be accurately approximated by SDEs. This finding opens a door to future modeling work using SDE techniques to further illuminate the effects of ion channel fluctuations on electrically active cells.
A new derivation of Akcasu's 'MLP' equations for 1-D particle transport in stochastic media
International Nuclear Information System (INIS)
Larsen, Edward W.; Prinja, Anil K.
2008-01-01
This paper presents a new derivation of Akcasu's modified Levermore-Pomraning (MLP) model, which estimates the ensemble-averaged angular flux for particle transport problems in 1-D geometrically random media. The significant new feature of the MLP equations is that, unlike the earlier Levermore-Pomraning (LP) model, the MLP equations are exact for certain classes of problems with scattering. We also show, via asymptotic analyses, that the MLP equations are accurate in the atomic mix and diffusion limits
A derivation of Akcasu's 'MLP' equations for 1-D particle transport in stochastic media
International Nuclear Information System (INIS)
Larsen, E. W.; Prinja, A. K.
2007-01-01
This paper presents a new derivation of Akcasu's Modified Levermore-Pomraning (MLP) model for estimating the ensemble-averaged angular flux for particle transport problems in 1-D geometrically random media. The significant new feature of the MLP equations is that, unlike the earlier Levermore-Pomraning (LP) model, the MLP equations are exact for certain classes of problems with scattering. (authors)
On the XFEL Schrödinger Equation: Highly Oscillatory Magnetic Potentials and Time Averaging
Antonelli, Paolo
2014-01-14
We analyse a nonlinear Schrödinger equation for the time-evolution of the wave function of an electron beam, interacting selfconsistently through a Hartree-Fock nonlinearity and through the repulsive Coulomb interaction of an atomic nucleus. The electrons are supposed to move under the action of a time dependent, rapidly periodically oscillating electromagnetic potential. This can be considered a simplified effective single particle model for an X-ray free electron laser. We prove the existence and uniqueness for the Cauchy problem and the convergence of wave-functions to corresponding solutions of a Schrödinger equation with a time-averaged Coulomb potential in the high frequency limit for the oscillations of the electromagnetic potential. © 2014 Springer-Verlag Berlin Heidelberg.
Barzel, Baruch; Biham, Ofer; Kupferman, Raz; Lipshtat, Azi; Zait, Amir
2010-08-01
Chemical reaction networks which exhibit strong fluctuations are common in microscopic systems in which reactants appear in low copy numbers. The analysis of these networks requires stochastic methods, which come in two forms: direct integration of the master equation and Monte Carlo simulations. The master equation becomes infeasible for large networks because the number of equations increases exponentially with the number of reactive species. Monte Carlo methods, which are more efficient in integrating over the exponentially large phase space, also become impractical due to the large amounts of noisy data that need to be stored and analyzed. The recently introduced multiplane method [A. Lipshtat and O. Biham, Phys. Rev. Lett. 93, 170601 (2004)10.1103/PhysRevLett.93.170601] is an efficient framework for the stochastic analysis of large reaction networks. It is a dimensional reduction method, based on the master equation, which provides a dramatic reduction in the number of equations without compromising the accuracy of the results. The reduction is achieved by breaking the network into a set of maximal fully connected subnetworks (maximal cliques). A separate master equation is written for the reduced probability distribution associated with each clique, with suitable coupling terms between them. This method is highly efficient in the case of sparse networks, in which the maximal cliques tend to be small. However, in dense networks some of the cliques may be rather large and the dimensional reduction is not as effective. Furthermore, the derivation of the multiplane equations from the master equation is tedious and difficult. Here we present the reduced-multiplane method in which the maximal cliques are broken down to the fundamental two-vertex cliques. The number of equations is further reduced, making the method highly efficient even for dense networks. Moreover, the equations take a simpler form, which can be easily constructed using a diagrammatic procedure
International Nuclear Information System (INIS)
Angstmann, C.N.; Donnelly, I.C.; Henry, B.I.; Jacobs, B.A.; Langlands, T.A.M.; Nichols, J.A.
2016-01-01
We have introduced a new explicit numerical method, based on a discrete stochastic process, for solving a class of fractional partial differential equations that model reaction subdiffusion. The scheme is derived from the master equations for the evolution of the probability density of a sum of discrete time random walks. We show that the diffusion limit of the master equations recovers the fractional partial differential equation of interest. This limiting procedure guarantees the consistency of the numerical scheme. The positivity of the solution and stability results are simply obtained, provided that the underlying process is well posed. We also show that the method can be applied to standard reaction–diffusion equations. This work highlights the broader applicability of using discrete stochastic processes to provide numerical schemes for partial differential equations, including fractional partial differential equations.
Nonlinear stochastic heat equations with cubic nonlinearities and additive Q-regular noise in R^1
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Henri Schurz
2010-09-01
Full Text Available Semilinear stochastic heat equations perturbed by cubic-type nonlinearities and additive space-time noise with homogeneous boundary conditions are discussed in R^1. The space-time noise is supposed to be Gaussian in time and possesses a Fourier expansion in space along the eigenfunctions of underlying Lapace operators. We follow the concept of approximate strong (classical Fourier solutions. The existence of unique continuous L^2-bounded solutions is proved. Furthermore, we present a procedure for its numerical approximation based on nonstandard methods (linear-implicit and justify their stability and consistency. The behavior of related total energy functional turns out to be crucial in the presented analysis.
Lima, L. S.; Miranda, L. L. B.
2018-01-01
We have used the Itô's stochastic differential equation for the double well with additive white noise as a mathematical model for price dynamics of the financial market. We have presented a model which allows us to test within the same framework the comparative explanatory power of rational agents versus irrational agents, with respect to the facts of financial markets. We have obtained the mean price in terms of the β parameter that represents the force of the randomness term of the model.
Tsallis distributions and 1/f noise from nonlinear stochastic differential equations.
Ruseckas, J; Kaulakys, B
2011-11-01
Probability distributions that emerge from the formalism of nonextensive statistical mechanics have been applied to a variety of problems. In this article we unite modeling of such distributions with the model of widespread 1/f noise. We propose a class of nonlinear stochastic differential equations giving both the q-exponential or q-Gaussian distributions of signal intensity, revealing long-range correlations and 1/f(β) behavior of the power spectral density. The superstatistical framework to get 1/f(β) noise with q-exponential and q-Gaussian distributions of the signal intensity is proposed, as well.
Ohkubo, Jun
2011-12-01
A scheme is developed for estimating state-dependent drift and diffusion coefficients in a stochastic differential equation from time-series data. The scheme does not require to specify parametric forms for the drift and diffusion coefficients in advance. In order to perform the nonparametric estimation, a maximum likelihood method is combined with a concept based on a kernel density estimation. In order to deal with discrete observation or sparsity of the time-series data, a local linearization method is employed, which enables a fast estimation.
Gundersen, Kenneth; Kvaløy, Jan Terje; Eftestøl, Trygve; Kramer-Johansen, Jo
2015-10-15
For patients undergoing cardiopulmonary resuscitation (CPR) and being in a shockable rhythm, the coarseness of the electrocardiogram (ECG) signal is an indicator of the state of the patient. In the current work, we show how mixed effects stochastic differential equations (SDE) models, commonly used in pharmacokinetic and pharmacodynamic modelling, can be used to model the relationship between CPR quality measurements and ECG coarseness. This is a novel application of mixed effects SDE models to a setting quite different from previous applications of such models and where using such models nicely solves many of the challenges involved in analysing the available data. Copyright © 2015 John Wiley & Sons, Ltd.
Random recurrence equations and ruin in a Markov-dependent stochastic economic environment
DEFF Research Database (Denmark)
Collamore, Jeffrey F.
2009-01-01
series models. Our results build upon work of Goldie, who has developed tail asymptotics applicable for independent sequences of random variables subject to a random recurrence equation. In contrast, we adopt a general approach based on the theory of Harris recurrent Markov chains and the associated......We develop sharp large deviation asymptotics for the probability of ruin in a Markov-dependent stochastic economic environment and study the extremes for some related Markovian processes which arise in financial and insurance mathematics, related to perpetuities and the ARCH(1) and GARCH(1,1) time...
International Nuclear Information System (INIS)
Frank, T.D.
2007-01-01
We present a generalized Kramers-Moyal expansion for stochastic differential equations with single and multiple delays. In particular, we show that the delay Fokker-Planck equation derived earlier in the literature is a special case of the proposed Kramers-Moyal expansion. Applications for bond pricing and a self-inhibitory neuron model are discussed
On the statistical mechanics of the 2D stochastic Euler equation
Bouchet, Freddy; Laurie, Jason; Zaboronski, Oleg
2011-12-01
The dynamics of vortices and large scale structures is qualitatively very different in two dimensional flows compared to its three dimensional counterparts, due to the presence of multiple integrals of motion. These are believed to be responsible for a variety of phenomena observed in Euler flow such as the formation of large scale coherent structures, the existence of meta-stable states and random abrupt changes in the topology of the flow. In this paper we study stochastic dynamics of the finite dimensional approximation of the 2D Euler flow based on Lie algebra su(N) which preserves all integrals of motion. In particular, we exploit rich algebraic structure responsible for the existence of Euler's conservation laws to calculate the invariant measures and explore their properties and also study the approach to equilibrium. Unexpectedly, we find deep connections between equilibrium measures of finite dimensional su(N) truncations of the stochastic Euler equations and random matrix models. Our work can be regarded as a preparation for addressing the questions of large scale structures, meta-stability and the dynamics of random transitions between different flow topologies in stochastic 2D Euler flows.
Higher-order stochastic differential equations and the positive Wigner function
Drummond, P. D.
2017-12-01
General higher-order stochastic processes that correspond to any diffusion-type tensor of higher than second order are obtained. The relationship of multivariate higher-order stochastic differential equations with tensor decomposition theory and tensor rank is explained. Techniques for generating the requisite complex higher-order noise are proved to exist either using polar coordinates and γ distributions, or from products of Gaussian variates. This method is shown to allow the calculation of the dynamics of the Wigner function, after it is extended to a complex phase space. The results are illustrated physically through dynamical calculations of the positive Wigner distribution for three-mode parametric downconversion, widely used in quantum optics. The approach eliminates paradoxes arising from truncation of the higher derivative terms in Wigner function time evolution. Anomalous results of negative populations and vacuum scattering found in truncated Wigner quantum simulations in quantum optics and Bose-Einstein condensate dynamics are shown not to occur with this type of stochastic theory.
Van Gorder, Robert A
2010-11-01
We apply the δ-expansion method to nonlinear stochastic differential equations describing wave propagation in a random medium. In particular, we focus our attention on a model describing a nonlinear wave propagating in a turbulent atmosphere which has random variations in the refractive index (we take these variations to be stochastic). The method allows us to construct much more reasonable perturbation solutions with relatively few terms (compared to standard "small-parameter" perturbation methods) due to more accurate linearization used in constructing the initial approximation. We demonstrate that the method allows one to compute effective wave numbers more precisely than other methods applied to the problem in the literature. The method also picks up on the stochastic damping of the solutions quickly, holding all of the relevant data in the initial term. These properties allow for both a qualitative and a quantitative construction of physically meaningful solutions. In particular, we show that the method allows one to retain higher-order harmonics which are hard to capture with standard perturbation methods based on small parameters.
Talmon, A.M.
1992-01-01
Flow equations for application in river bends are derived. Coordinate transformations are achieved by tensor calculus. The momentum equations are formulated in a cylindrical coordinate system and in a cylindrical bed-following coordinate system. The depth-averaged momentum equations in a cylindrical
Numerical solution of stochastic differential equations with Poisson and Lévy white noise
Grigoriu, M.
2009-08-01
A fixed time step method is developed for integrating stochastic differential equations (SDE’s) with Poisson white noise (PWN) and Lévy white noise (LWN). The method for integrating SDE’s with PWN has the same structure as that proposed by Kim [Phys. Rev. E 76, 011109 (2007)], but is established by using different arguments. The integration of SDE’s with LWN is based on a representation of Lévy processes by sums of scaled Brownian motions and compound Poisson processes. It is shown that the numerical solutions of SDE’s with PWN and LWN converge weakly to the exact solutions of these equations, so that they can be used to estimate not only marginal properties but also distributions of functionals of the exact solutions. Numerical examples are used to demonstrate the applications and the accuracy of the proposed integration algorithms.
Directory of Open Access Journals (Sweden)
Fakhrodin Mohammadi
2017-10-01
Full Text Available Stochastic fractional differential equations (SFDEs have been used for modeling many physical problems in the fields of turbulance, heterogeneous, flows and matrials, viscoelasticity and electromagnetic theory. In this paper, an efficient wavelet Galerkin method based on the second kind Chebyshev wavelets are proposed for approximate solution of SFDEs. In this approach, operational matrices of the second kind Chebyshev wavelets are used for reducing SFDEs to a linear system of algebraic equations that can be solved easily. Convergence and error analysis of the proposed method is considered. Some numerical examples are performed to confirm the applicability and efficiency of the proposed method.
Numerical solution of stochastic differential equations with Poisson and Lévy white noise.
Grigoriu, M
2009-08-01
A fixed time step method is developed for integrating stochastic differential equations (SDE's) with Poisson white noise (PWN) and Lévy white noise (LWN). The method for integrating SDE's with PWN has the same structure as that proposed by Kim [Phys. Rev. E 76, 011109 (2007)], but is established by using different arguments. The integration of SDE's with LWN is based on a representation of Lévy processes by sums of scaled Brownian motions and compound Poisson processes. It is shown that the numerical solutions of SDE's with PWN and LWN converge weakly to the exact solutions of these equations, so that they can be used to estimate not only marginal properties but also distributions of functionals of the exact solutions. Numerical examples are used to demonstrate the applications and the accuracy of the proposed integration algorithms.
Quasi-linear PDEs and forward-backward stochastic differential equations: Weak solutions
Feng, Chunrong; Wang, Xince; Zhao, Huaizhong
2018-01-01
In this paper, we study the existence, uniqueness and the probabilistic representation of the weak solutions of quasi-linear parabolic and elliptic partial differential equations (PDEs) in the Sobolev space Hρ1 (Rd). For this, we study first the solutions of forward-backward stochastic differential equations (FBSDEs) with smooth coefficients, regularity of solutions and their connection with classical solutions of quasi-linear parabolic PDEs. Then using the approximation procedure, we establish their convergence in the Sobolev space to the solutions of the FBSDES in the space Lρ2 (Rd ;Rd) ⊗ Lρ2 (Rd ;Rk) ⊗ Lρ2 (Rd ;R k × d). This gives a connection with the weak solutions of quasi-linear parabolic PDEs. Finally, we study the unique weak solutions of quasi-linear elliptic PDEs using the solutions of the FBSDEs on infinite horizon.
XMDS2: Fast, scalable simulation of coupled stochastic partial differential equations
Dennis, Graham R.; Hope, Joseph J.; Johnsson, Mattias T.
2013-01-01
XMDS2 is a cross-platform, GPL-licensed, open source package for numerically integrating initial value problems that range from a single ordinary differential equation up to systems of coupled stochastic partial differential equations. The equations are described in a high-level XML-based script, and the package generates low-level optionally parallelised C++ code for the efficient solution of those equations. It combines the advantages of high-level simulations, namely fast and low-error development, with the speed, portability and scalability of hand-written code. XMDS2 is a complete redesign of the XMDS package, and features support for a much wider problem space while also producing faster code. Program summaryProgram title: XMDS2 Catalogue identifier: AENK_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AENK_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: GNU General Public License, version 2 No. of lines in distributed program, including test data, etc.: 872490 No. of bytes in distributed program, including test data, etc.: 45522370 Distribution format: tar.gz Programming language: Python and C++. Computer: Any computer with a Unix-like system, a C++ compiler and Python. Operating system: Any Unix-like system; developed under Mac OS X and GNU/Linux. RAM: Problem dependent (roughly 50 bytes per grid point) Classification: 4.3, 6.5. External routines: The external libraries required are problem-dependent. Uses FFTW3 Fourier transforms (used only for FFT-based spectral methods), dSFMT random number generation (used only for stochastic problems), MPI message-passing interface (used only for distributed problems), HDF5, GNU Scientific Library (used only for Bessel-based spectral methods) and a BLAS implementation (used only for non-FFT-based spectral methods). Nature of problem: General coupled initial-value stochastic partial differential equations. Solution method: Spectral method
Pettersson, Per; Nordström, Jan; Doostan, Alireza
2016-02-01
We present a well-posed stochastic Galerkin formulation of the incompressible Navier-Stokes equations with uncertainty in model parameters or the initial and boundary conditions. The stochastic Galerkin method involves representation of the solution through generalized polynomial chaos expansion and projection of the governing equations onto stochastic basis functions, resulting in an extended system of equations. A relatively low-order generalized polynomial chaos expansion is sufficient to capture the stochastic solution for the problem considered. We derive boundary conditions for the continuous form of the stochastic Galerkin formulation of the velocity and pressure equations. The resulting problem formulation leads to an energy estimate for the divergence. With suitable boundary data on the pressure and velocity, the energy estimate implies zero divergence of the velocity field. Based on the analysis of the continuous equations, we present a semi-discretized system where the spatial derivatives are approximated using finite difference operators with a summation-by-parts property. With a suitable choice of dissipative boundary conditions imposed weakly through penalty terms, the semi-discrete scheme is shown to be stable. Numerical experiments in the laminar flow regime corroborate the theoretical results and we obtain high-order accurate results for the solution variables and the velocity divergence converges to zero as the mesh is refined.
International Nuclear Information System (INIS)
Gupta, Chinmaya; López, José Manuel; Azencott, Robert; Ott, William; Bennett, Matthew R.; Josić, Krešimir
2014-01-01
Delay is an important and ubiquitous aspect of many biochemical processes. For example, delay plays a central role in the dynamics of genetic regulatory networks as it stems from the sequential assembly of first mRNA and then protein. Genetic regulatory networks are therefore frequently modeled as stochastic birth-death processes with delay. Here, we examine the relationship between delay birth-death processes and their appropriate approximating delay chemical Langevin equations. We prove a quantitative bound on the error between the pathwise realizations of these two processes. Our results hold for both fixed delay and distributed delay. Simulations demonstrate that the delay chemical Langevin approximation is accurate even at moderate system sizes. It captures dynamical features such as the oscillatory behavior in negative feedback circuits, cross-correlations between nodes in a network, and spatial and temporal information in two commonly studied motifs of metastability in biochemical systems. Overall, these results provide a foundation for using delay stochastic differential equations to approximate the dynamics of birth-death processes with delay
Gupta, Chinmaya; López, José Manuel; Azencott, Robert; Bennett, Matthew R; Josić, Krešimir; Ott, William
2014-05-28
Delay is an important and ubiquitous aspect of many biochemical processes. For example, delay plays a central role in the dynamics of genetic regulatory networks as it stems from the sequential assembly of first mRNA and then protein. Genetic regulatory networks are therefore frequently modeled as stochastic birth-death processes with delay. Here, we examine the relationship between delay birth-death processes and their appropriate approximating delay chemical Langevin equations. We prove a quantitative bound on the error between the pathwise realizations of these two processes. Our results hold for both fixed delay and distributed delay. Simulations demonstrate that the delay chemical Langevin approximation is accurate even at moderate system sizes. It captures dynamical features such as the oscillatory behavior in negative feedback circuits, cross-correlations between nodes in a network, and spatial and temporal information in two commonly studied motifs of metastability in biochemical systems. Overall, these results provide a foundation for using delay stochastic differential equations to approximate the dynamics of birth-death processes with delay.
Energy Technology Data Exchange (ETDEWEB)
Ceschia, M.; Garfagnini, R.; Lavenia, A.; Toppano, E.
The daily average air temperatures collected during 1984 in Udine and Lignano are studied using stochastic processes. In order to obtain the seasonal component, a computational method has been developed involving the Fourier expansion, while the study of the stationary residual series is carried out by means of ARMA recursive filters. It was found that an autoregressive model of the third order represents the residual data fairly well. This method may also be applied to other meteorological parameters with significant frequencies and with a time behaviour which can be made stationary at least in the wide sense. This work was carried out in the scope of the MICMET program which is aimed at obtaining a detailed analysis of micrometeorological phenomena in the region of Friuli-Venezia Giulia.
Energy Technology Data Exchange (ETDEWEB)
Cui, Jianbo, E-mail: jianbocui@lsec.cc.ac.cn [Institute of Computational Mathematics and Scientific/Engineering Computing, Chinese Academy of Sciences, Beijing, 100190 (China); Hong, Jialin, E-mail: hjl@lsec.cc.ac.cn [Institute of Computational Mathematics and Scientific/Engineering Computing, Chinese Academy of Sciences, Beijing, 100190 (China); Liu, Zhihui, E-mail: liuzhihui@lsec.cc.ac.cn [Institute of Computational Mathematics and Scientific/Engineering Computing, Chinese Academy of Sciences, Beijing, 100190 (China); Zhou, Weien, E-mail: weienzhou@nudt.edu.cn [College of Science, National University of Defense Technology, Changsha 410073 (China)
2017-08-01
We indicate that the nonlinear Schrödinger equation with white noise dispersion possesses stochastic symplectic and multi-symplectic structures. Based on these structures, we propose the stochastic symplectic and multi-symplectic methods, which preserve the continuous and discrete charge conservation laws, respectively. Moreover, we show that the proposed methods are convergent with temporal order one in probability. Numerical experiments are presented to verify our theoretical results.
International Nuclear Information System (INIS)
Cui, Jianbo; Hong, Jialin; Liu, Zhihui; Zhou, Weien
2017-01-01
We indicate that the nonlinear Schrödinger equation with white noise dispersion possesses stochastic symplectic and multi-symplectic structures. Based on these structures, we propose the stochastic symplectic and multi-symplectic methods, which preserve the continuous and discrete charge conservation laws, respectively. Moreover, we show that the proposed methods are convergent with temporal order one in probability. Numerical experiments are presented to verify our theoretical results.
Periodic solutions of Lienard differential equations via averaging theory of order two
Directory of Open Access Journals (Sweden)
JAUME LLIBRE
2015-12-01
Full Text Available Abstract For ε ≠ 0sufficiently small we provide sufficient conditions for the existence of periodic solutions for the Lienard differential equations of the form x ′′ + f ( x x ′ + n 2 x + g ( x = ε 2 p 1 ( t + ε 3 p 2 ( t , where n is a positive integer, f : ℝ → ℝis a C 3function, g : ℝ → ℝis a C 4function, and p i : ℝ → ℝfor i = 1 , 2are continuous 2 π–periodic function. The main tool used in this paper is the averaging theory of second order. We also provide one application of the main result obtained.
Periodic solutions of Lienard differential equations via averaging theory of order two.
Llibre, Jaume; Novaes, Douglas D; Teixeira, Marco A
2015-01-01
For ε ≠ 0 sufficiently small we provide sufficient conditions for the existence of periodic solutions for the Lienard differential equations of the form x'' + f (x) x' + n2x + g (x) = ε2p1 (t) + ε3 p2(t), where n is a positive integer, f : ℝ → ℝ is a C 3 function, g : ℝ → ℝ is a C 4 function, and p i : ℝ → ℝ for i = 1, 2 are continuous 2π-periodic function. The main tool used in this paper is the averaging theory of second order. We also provide one application of the main result obtained.
International Nuclear Information System (INIS)
Espinosa-Paredes, Gilberto
2010-01-01
The aim of this paper is to propose a framework to obtain a new formulation for multiphase flow conservation equations without length-scale restrictions, based on the non-local form of the averaged volume conservation equations. The simplification of the local averaging volume of the conservation equations to obtain practical equations is subject to the following length-scale restrictions: d << l << L, where d is the characteristic length of the dispersed phases, l is the characteristic length of the averaging volume, and L is the characteristic length of the physical system. If the foregoing inequality does not hold, or if the scale of the problem of interest is of the order of l, the averaging technique and therefore, the macroscopic theories of multiphase flow should be modified in order to include appropriate considerations and terms in the corresponding equations. In these cases the local form of the averaged volume conservation equations are not appropriate to describe the multiphase system. As an example of the conservation equations without length-scale restrictions, the natural circulation boiling water reactor was consider to study the non-local effects on the thermal-hydraulic core performance during steady-state and transient behaviors, and the results were compared with the classic local averaging volume conservation equations.
Scaling Relations and Self-Similarity of 3-Dimensional Reynolds-Averaged Navier-Stokes Equations.
Ercan, Ali; Kavvas, M Levent
2017-07-25
Scaling conditions to achieve self-similar solutions of 3-Dimensional (3D) Reynolds-Averaged Navier-Stokes Equations, as an initial and boundary value problem, are obtained by utilizing Lie Group of Point Scaling Transformations. By means of an open-source Navier-Stokes solver and the derived self-similarity conditions, we demonstrated self-similarity within the time variation of flow dynamics for a rigid-lid cavity problem under both up-scaled and down-scaled domains. The strength of the proposed approach lies in its ability to consider the underlying flow dynamics through not only from the governing equations under consideration but also from the initial and boundary conditions, hence allowing to obtain perfect self-similarity in different time and space scales. The proposed methodology can be a valuable tool in obtaining self-similar flow dynamics under preferred level of detail, which can be represented by initial and boundary value problems under specific assumptions.
Model identification using stochastic differential equation grey-box models in diabetes.
Duun-Henriksen, Anne Katrine; Schmidt, Signe; Røge, Rikke Meldgaard; Møller, Jonas Bech; Nørgaard, Kirsten; Jørgensen, John Bagterp; Madsen, Henrik
2013-03-01
The acceptance of virtual preclinical testing of control algorithms is growing and thus also the need for robust and reliable models. Models based on ordinary differential equations (ODEs) can rarely be validated with standard statistical tools. Stochastic differential equations (SDEs) offer the possibility of building models that can be validated statistically and that are capable of predicting not only a realistic trajectory, but also the uncertainty of the prediction. In an SDE, the prediction error is split into two noise terms. This separation ensures that the errors are uncorrelated and provides the possibility to pinpoint model deficiencies. An identifiable model of the glucoregulatory system in a type 1 diabetes mellitus (T1DM) patient is used as the basis for development of a stochastic-differential-equation-based grey-box model (SDE-GB). The parameters are estimated on clinical data from four T1DM patients. The optimal SDE-GB is determined from likelihood-ratio tests. Finally, parameter tracking is used to track the variation in the "time to peak of meal response" parameter. We found that the transformation of the ODE model into an SDE-GB resulted in a significant improvement in the prediction and uncorrelated errors. Tracking of the "peak time of meal absorption" parameter showed that the absorption rate varied according to meal type. This study shows the potential of using SDE-GBs in diabetes modeling. Improved model predictions were obtained due to the separation of the prediction error. SDE-GBs offer a solid framework for using statistical tools for model validation and model development. © 2013 Diabetes Technology Society.
Leander, Jacob; Almquist, Joachim; Ahlström, Christine; Gabrielsson, Johan; Jirstrand, Mats
2015-05-01
Inclusion of stochastic differential equations in mixed effects models provides means to quantify and distinguish three sources of variability in data. In addition to the two commonly encountered sources, measurement error and interindividual variability, we also consider uncertainty in the dynamical model itself. To this end, we extend the ordinary differential equation setting used in nonlinear mixed effects models to include stochastic differential equations. The approximate population likelihood is derived using the first-order conditional estimation with interaction method and extended Kalman filtering. To illustrate the application of the stochastic differential mixed effects model, two pharmacokinetic models are considered. First, we use a stochastic one-compartmental model with first-order input and nonlinear elimination to generate synthetic data in a simulated study. We show that by using the proposed method, the three sources of variability can be successfully separated. If the stochastic part is neglected, the parameter estimates become biased, and the measurement error variance is significantly overestimated. Second, we consider an extension to a stochastic pharmacokinetic model in a preclinical study of nicotinic acid kinetics in obese Zucker rats. The parameter estimates are compared between a deterministic and a stochastic NiAc disposition model, respectively. Discrepancies between model predictions and observations, previously described as measurement noise only, are now separated into a comparatively lower level of measurement noise and a significant uncertainty in model dynamics. These examples demonstrate that stochastic differential mixed effects models are useful tools for identifying incomplete or inaccurate model dynamics and for reducing potential bias in parameter estimates due to such model deficiencies.
A new nonlinear turbulence model based on Partially-Averaged Navier-Stokes Equations
International Nuclear Information System (INIS)
Liu, J T; Wu, Y L; Cai, C; Liu, S H; Wang, L Q
2013-01-01
Partially-averaged Navier-Stokes (PANS) Model was recognized as a Reynolds-averaged Navier-Stokes (RANS) to direct numerical simulation (DNS) bridging method. PANS model was purported for any filter width-from RANS to DNS. PANS method also shared some similarities with the currently popular URANS (unsteady RANS) method. In this paper, a new PANS model was proposed, which was based on RNG k-ε turbulence model. The Standard and RNG k-ε turbulence model were both isotropic models, as well as PANS models. The sheer stress in those PANS models was solved by linear equation. The linear hypothesis was not accurate in the simulation of complex flow, such as stall phenomenon. The sheer stress here was solved by nonlinear method proposed by Ehrhard. Then, the nonlinear PANS model was set up. The pressure coefficient of the suction side of the NACA0015 hydrofoil was predicted. The result of pressure coefficient agrees well with experimental result, which proves that the nonlinear PANS model can capture the high pressure gradient flow. A low specific centrifugal pump was used to verify the capacity of the nonlinear PANS model. The comparison between the simulation results of the centrifugal pump and Particle Image Velocimetry (PIV) results proves that the nonlinear PANS model can be used in the prediction of complex flow field
Efficient Multilevel and Multi-index Sampling Methods in Stochastic Differential Equations
Haji-Ali, Abdul Lateef
2016-05-22
of this thesis is the novel Multi-index Monte Carlo (MIMC) method which is an extension of MLMC in high dimensional problems with significant computational savings. Under reasonable assumptions on the weak and variance convergence, which are related to the mixed regularity of the underlying problem and the discretization method, the order of the computational complexity of MIMC is, at worst up to a logarithmic factor, independent of the dimensionality of the underlying parametric equation. We also apply the same multi-index methodology to another sampling method, namely the Stochastic Collocation method. Hence, the novel Multi-index Stochastic Collocation method is proposed and is shown to be more efficient in problems with sufficient mixed regularity than our novel MIMC method and other standard methods. Finally, MIMC is applied to approximate quantities of interest of stochastic particle systems in the mean-field when the number of particles tends to infinity. To approximate these quantities of interest up to an error tolerance, TOL, MIMC has a computational complexity of O(TOL-2log(TOL)2). This complexity is achieved by building a hierarchy based on two discretization parameters: the number of time steps in an Milstein scheme and the number of particles in the particle system. Moreover, we use a partitioning estimator to increase the correlation between two stochastic particle systems with different sizes. In comparison, the optimal computational complexity of MLMC in this case is O(TOL-3) and the computational complexity of Monte Carlo is O(TOL-4).
Figueredo, Grazziela P; Siebers, Peer-Olaf; Owen, Markus R; Reps, Jenna; Aickelin, Uwe
2014-01-01
There is great potential to be explored regarding the use of agent-based modelling and simulation as an alternative paradigm to investigate early-stage cancer interactions with the immune system. It does not suffer from some limitations of ordinary differential equation models, such as the lack of stochasticity, representation of individual behaviours rather than aggregates and individual memory. In this paper we investigate the potential contribution of agent-based modelling and simulation when contrasted with stochastic versions of ODE models using early-stage cancer examples. We seek answers to the following questions: (1) Does this new stochastic formulation produce similar results to the agent-based version? (2) Can these methods be used interchangeably? (3) Do agent-based models outcomes reveal any benefit when compared to the Gillespie results? To answer these research questions we investigate three well-established mathematical models describing interactions between tumour cells and immune elements. These case studies were re-conceptualised under an agent-based perspective and also converted to the Gillespie algorithm formulation. Our interest in this work, therefore, is to establish a methodological discussion regarding the usability of different simulation approaches, rather than provide further biological insights into the investigated case studies. Our results show that it is possible to obtain equivalent models that implement the same mechanisms; however, the incapacity of the Gillespie algorithm to retain individual memory of past events affects the similarity of some results. Furthermore, the emergent behaviour of ABMS produces extra patters of behaviour in the system, which was not obtained by the Gillespie algorithm.
Pitchfork and Hopf bifurcation thresholds in stochastic equations with delayed feedback
Gaudreault, Mathieu; Lépine, Françoise; Viñals, Jorge
2009-12-01
The bifurcation diagram of a model stochastic differential equation with delayed feedback is presented. We are motivated by recent research on stochastic effects in models of transcriptional gene regulation. We start from the normal form for a pitchfork bifurcation, and add multiplicative or parametric noise and linear delayed feedback. The latter is sufficient to originate a Hopf bifurcation in that region of parameters in which there is a sufficiently strong negative feedback. We find a sharp bifurcation in parameter space, and define the threshold as the point in which the stationary distribution function p(x) changes from a delta function at the trivial state x=0 to p(x)˜xα at small x (with α=-1 exactly at threshold). We find that the bifurcation threshold is shifted by fluctuations relative to the deterministic limit by an amount that scales linearly with the noise intensity. Analytic calculations of the bifurcation threshold are also presented in the limit of small delay τ→0 that compare quite favorably with the numerical solutions even for moderate values of τ .
Stochastic Differential Equation Models for the Price of European CO2 Emissions Allowances
Directory of Open Access Journals (Sweden)
Wugan Cai
2017-02-01
Full Text Available Understanding the stochastic nature of emissions allowances is crucial for risk management in emissions trading markets. In this study, we discuss the emissions allowances spot price within the European Union Emissions Trading Scheme: Powernext and European Climate Exchange. To compare the fitness of five stochastic differential equations (SDEs to the European Union allowances spot price, we apply regression theory to obtain the point and interval estimations for the parameters of the SDEs. An empirical evaluation demonstrates that the mean reverting square root process (MRSRP has the best fitness of five SDEs to forecast the spot price. To reduce the degree of smog, we develop a new trading scheme in which firms have to hand many more allowances to the government when they emit one unit of air pollution on heavy pollution days, versus one allowance on clean days. Thus, we set up the SDE MRSRP model with Markovian switching to analyse the evolution of the spot price in such a scheme. The analysis shows that the allowances spot price will not jump too much in the new scheme. The findings of this study could contribute to developing a new type of emissions trading.
Directory of Open Access Journals (Sweden)
Roberts Jason A
2006-01-01
Full Text Available We consider the reliability of some numerical methods in preserving the stability properties of the linear stochastic functional differential equation , where α, β, σ, τ ≥ 0 are real constants, and W(t is a standard Wiener process. The areas of the regions of asymptotic stability for the class of methods considered, indicated by the sufficient conditions for the discrete system, are shown to be equal in size to each other and we show that an upper bound can be put on the time-step parameter for the numerical method for which the system is asymptotically mean-square stable. We illustrate our results by means of numerical experiments and various stability diagrams. We examine the extent to which the continuous system can tolerate stochastic perturbations before losing its stability properties and we illustrate how one may accurately choose a numerical method to preserve the stability properties of the original problem in the numerical solution. Our numerical experiments also indicate that the quality of the sufficient conditions is very high.
Sanz, Luis; Alonso, Juan Antonio
2017-12-01
In this work we develop approximate aggregation techniques in the context of slow-fast linear population models governed by stochastic differential equations and apply the results to the treatment of populations with spatial heterogeneity. Approximate aggregation techniques allow one to transform a complex system involving many coupled variables and in which there are processes with different time scales, by a simpler reduced model with a fewer number of 'global' variables, in such a way that the dynamics of the former can be approximated by that of the latter. In our model we contemplate a linear fast deterministic process together with a linear slow process in which the parameters are affected by additive noise, and give conditions for the solutions corresponding to positive initial conditions to remain positive for all times. By letting the fast process reach equilibrium we build a reduced system with a lesser number of variables, and provide results relating the asymptotic behaviour of the first- and second-order moments of the population vector for the original and the reduced system. The general technique is illustrated by analysing a multiregional stochastic system in which dispersal is deterministic and the rate growth of the populations in each patch is affected by additive noise.
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data
Babuška, Ivo
2010-01-01
This work proposes and analyzes a stochastic collocation method for solving elliptic partial differential equations with random coefficients and forcing terms. These input data are assumed to depend on a finite number of random variables. The method consists of a Galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space, and naturally leads to the solution of uncoupled deterministic problems as in the Monte Carlo approach. It treats easily a wide range of situations, such as input data that depend nonlinearly on the random variables, diffusivity coefficients with unbounded second moments, and random variables that are correlated or even unbounded. We provide a rigorous convergence analysis and demonstrate exponential convergence of the “probability error” with respect to the number of Gauss points in each direction of the probability space, under some regularity assumptions on the random input data. Numerical examples show the effectiveness of the method. Finally, we include a section with developments posterior to the original publication of this work. There we review sparse grid stochastic collocation methods, which are effective collocation strategies for problems that depend on a moderately large number of random variables.
Moriguchi, Kai
2018-01-01
Existing growth equations have been derived by hypothesizing the growth rate as self-referencing. Although that hypothesis is appropriate for quantities with growth directly restricted to themselves, it requires the solution of differential equations that are difficult to be derived. However, for quantities of growth that are indirectly related to themselves, and for quantities that exhibit cumulative growth, such as growth of diameter or volume of trees with lignification, ordinary growth models with assumptions that include self-referencing or implied catabolism terms might not be appropriate. For such quantities, the author proposes an approach for derivation of growth models based on stochastic and microscopic interpretation. Results show that ordinary growth models can be interpreted from the perspective. The approach enables one to derive growth functions without solving differential equations. Based on that approach, growth functions are derived by integrating cumulative distribution functions. Three growth functions are generated using reasonable probability distributions. The fitness of the generated growth functions for real diameter growth data was compared with that of generalized ordinary growth models. Results show that the presented approach has high ability to generate growth models that fit data much better than ordinary growth models for a given number of parameters.
Hall, Eric Joseph
2016-12-08
We derive computable error estimates for finite element approximations of linear elliptic partial differential equations with rough stochastic coefficients. In this setting, the exact solutions contain high frequency content that standard a posteriori error estimates fail to capture. We propose goal-oriented estimates, based on local error indicators, for the pathwise Galerkin and expected quadrature errors committed in standard, continuous, piecewise linear finite element approximations. Derived using easily validated assumptions, these novel estimates can be computed at a relatively low cost and have applications to subsurface flow problems in geophysics where the conductivities are assumed to have lognormal distributions with low regularity. Our theory is supported by numerical experiments on test problems in one and two dimensions.
Zalys-Geller, E.; Hatridge, M.; Silveri, M.; Narla, A.; Sliwa, K. M.; Shankar, S.; Girvin, S. M.; Devoret, M. H.
2015-03-01
Remote entanglement of two superconducting qubits may be accomplished by first entangling them with flying coherent microwave pulses, and then erasing the which-path information of these pulses by using a non-degenerate parametric amplifier such as the Josephson Parametric Converter (JPC). Crucially, this process requires no direct interaction between the two qubits. The JPC, however, will fail to completely erase the which-path information if the flying microwave pulses encode any difference in dynamics of the two qubit-cavity systems. This which-path information can easily arise from mismatches in the cavity linewidths and the cavity dispersive shifts from their respective qubits. Through analysis of the Stochastic Master Equation for this system, we have found a strategy for shaping the measurement pulses to eliminate the effect of these mismatches on the entangling measurement. We have then confirmed the effectiveness of this strategy by numerical simulation. Work supported by: IARPA, ARO, and NSF.
Modelling the heat dynamics of a building using stochastic differential equations
DEFF Research Database (Denmark)
Andersen, Klaus Kaae; Madsen, Henrik; Hansen, Lars Henrik
2000-01-01
This paper describes the continuous time modelling of the heat dynamics of a building. The considered building is a residential like test house divided into two test rooms with a water based central heating. Each test room is divided into thermal zones in order to describe both short and long term...... variations. Besides modelling the heat transfer between thermal zones, attention is put on modelling the heat input from radiators and solar radiation. The applied modelling procedure is based on collected building performance data and statistical methods. The statistical methods are used in parameter...... estimation and model validation, while physical knowledge is used in forming the model structure. The suggested lumped parameter model is thus based on thermodynamics and formulated as a system of stochastic differential equations. Due to the continuous time formulation the parameters of the model...
Tail estimates for stochastic fixed point equations via nonlinear renewal theory
DEFF Research Database (Denmark)
Collamore, Jeffrey F.; Vidyashankar, Anand N.
2013-01-01
estimate P(V>u)~Cu^{-r} as u tends to infinity, and also present a corresponding Lundberg-type upper bound. To this end, we introduce a novel dual change of measure on a random time interval and analyze the path properties, using nonlinear renewal theory, of the Markov chain resulting from the forward...... of our methods to Markov-driven processes.......This paper presents precise large deviation estimates for solutions to stochastic fixed point equations of the type V =_D f(V), where f(v)=Av+g(v) for a random function g(v)=o(v) a.s. as v tends to infinity. Specifically, we provide an explicit characterization of the pair (C,r) in the tail...
Kinetics of subdiffusion-assisted reactions: non-Markovian stochastic Liouville equation approach
International Nuclear Information System (INIS)
Shushin, A I
2005-01-01
Anomalous specific features of the kinetics of subdiffusion-assisted bimolecular reactions (time-dependence, dependence on parameters of systems, etc) are analysed in detail with the use of the non-Markovian stochastic Liouville equation (SLE), which has been recently derived within the continuous-time random-walk (CTRW) approach. In the CTRW approach, subdiffusive motion of particles is modelled by jumps whose onset probability distribution function is of a long-tailed form. The non-Markovian SLE allows for rigorous describing of some peculiarities of these reactions; for example, very slow long-time behaviour of the kinetics, non-analytical dependence of the reaction rate on the reactivity of particles, strong manifestation of fluctuation kinetics showing itself in very slowly decreasing behaviour of the kinetics at very long times, etc
International Nuclear Information System (INIS)
Di Nunno, Giulia; Khedher, Asma; Vanmaele, Michèle
2015-01-01
We consider a backward stochastic differential equation with jumps (BSDEJ) which is driven by a Brownian motion and a Poisson random measure. We present two candidate-approximations to this BSDEJ and we prove that the solution of each candidate-approximation converges to the solution of the original BSDEJ in a space which we specify. We use this result to investigate in further detail the consequences of the choice of the model to (partial) hedging in incomplete markets in finance. As an application, we consider models in which the small variations in the price dynamics are modeled with a Poisson random measure with infinite activity and models in which these small variations are modeled with a Brownian motion or are cut off. Using the convergence results on BSDEJs, we show that quadratic hedging strategies are robust towards the approximation of the market prices and we derive an estimation of the model risk
Energy Technology Data Exchange (ETDEWEB)
Di Nunno, Giulia, E-mail: giulian@math.uio.no [University of Oslo, Center of Mathematics for Applications (Norway); Khedher, Asma, E-mail: asma.khedher@tum.de [Technische Universität München, Chair of Mathematical Finance (Germany); Vanmaele, Michèle, E-mail: michele.vanmaele@ugent.be [Ghent University, Department of Applied Mathematics, Computer Science and Statistics (Belgium)
2015-12-15
We consider a backward stochastic differential equation with jumps (BSDEJ) which is driven by a Brownian motion and a Poisson random measure. We present two candidate-approximations to this BSDEJ and we prove that the solution of each candidate-approximation converges to the solution of the original BSDEJ in a space which we specify. We use this result to investigate in further detail the consequences of the choice of the model to (partial) hedging in incomplete markets in finance. As an application, we consider models in which the small variations in the price dynamics are modeled with a Poisson random measure with infinite activity and models in which these small variations are modeled with a Brownian motion or are cut off. Using the convergence results on BSDEJs, we show that quadratic hedging strategies are robust towards the approximation of the market prices and we derive an estimation of the model risk.
DEFF Research Database (Denmark)
Knudsen, Thomas Skov
1997-01-01
Lipschitz continuous with respect to b when the space of drift coefficients is the set of Lipschitz continuous functions of sublinear growth endowed with the sup-norm. Furthermore, it is shown that this result does not hold if we interchange the role of b and c. However for SDEs where the coefficient vector......For stochastic differential equations (SDEs) of the form dX(t) = b(X)(t)) dt + sigma(X(t))dW(t) where b and sigma are Lipschitz continuous, it is shown that if we consider a fixed sigma is an element of C-5, bounded and with bounded derivatives, the random field of solutions is pathwise locally...
Zhang, Ling
2017-01-01
The main purpose of this paper is to investigate the strong convergence and exponential stability in mean square of the exponential Euler method to semi-linear stochastic delay differential equations (SLSDDEs). It is proved that the exponential Euler approximation solution converges to the analytic solution with the strong order [Formula: see text] to SLSDDEs. On the one hand, the classical stability theorem to SLSDDEs is given by the Lyapunov functions. However, in this paper we study the exponential stability in mean square of the exact solution to SLSDDEs by using the definition of logarithmic norm. On the other hand, the implicit Euler scheme to SLSDDEs is known to be exponentially stable in mean square for any step size. However, in this article we propose an explicit method to show that the exponential Euler method to SLSDDEs is proved to share the same stability for any step size by the property of logarithmic norm.
Xie, Bin
2018-01-01
In this paper, the main topic is to investigate the intermittent property of the one-dimensional stochastic heat equation driven by an inhomogeneous Brownian sheet, which is a noise deduced from the study of the catalytic super-Brownian motion. Under some proper conditions on the catalytic measure of the inhomogeneous Brownian sheet, we show that the solution is weakly full intermittent based on the estimates of moments of the solution. In particular, it is proved that the second moment of the solution grows at the exponential rate. The novelty is that the catalytic measure relative to the inhomogeneous noise is not required to be absolutely continuous with respect to the Lebesgue measure on R.
Gerencsér, Máté; Jentzen, Arnulf; Salimova, Diyora
2017-11-01
In a recent article (Jentzen et al. 2016 Commun. Math. Sci. 14 , 1477-1500 (doi:10.4310/CMS.2016.v14.n6.a1)), it has been established that, for every arbitrarily slow convergence speed and every natural number d ∈{4,5,…}, there exist d -dimensional stochastic differential equations with infinitely often differentiable and globally bounded coefficients such that no approximation method based on finitely many observations of the driving Brownian motion can converge in absolute mean to the solution faster than the given speed of convergence. In this paper, we strengthen the above result by proving that this slow convergence phenomenon also arises in two ( d =2) and three ( d =3) space dimensions.
A stochastic differential equation model for the foraging behavior of fish schools.
Tạ, Tôn Việt; Nguyen, Linh Thi Hoai
2018-03-15
Constructing models of living organisms locating food sources has important implications for understanding animal behavior and for the development of distribution technologies. This paper presents a novel simple model of stochastic differential equations for the foraging behavior of fish schools in a space including obstacles. The model is studied numerically. Three configurations of space with various food locations are considered. In the first configuration, fish swim in free but limited space. All individuals can find food with large probability while keeping their school structure. In the second and third configurations, they move in limited space with one and two obstacles, respectively. Our results reveal that the probability of foraging success is highest in the first configuration, and smallest in the third one. Furthermore, when school size increases up to an optimal value, the probability of foraging success tends to increase. When it exceeds an optimal value, the probability tends to decrease. The results agree with experimental observations.
Intermittency in relation with 1/f noise and stochastic differential equations.
Ruseckas, J; Kaulakys, B
2013-06-01
One of the models of intermittency is on-off intermittency, arising due to time-dependent forcing of a bifurcation parameter through a bifurcation point. For on-off intermittency, the power spectral density (PSD) of the time-dependent deviation from the invariant subspace in a low frequency region exhibits 1/√f power-law noise. Here, we investigate a mechanism of intermittency, similar to the on-off intermittency, occurring in nonlinear dynamical systems with invariant subspace. In contrast to the on-off intermittency, we consider the case where the transverse Lyapunov exponent is zero. We show that for such nonlinear dynamical systems, the power spectral density of the deviation from the invariant subspace can have 1/f(β) form in a wide range of frequencies. That is, such nonlinear systems exhibit 1/f noise. The connection with the stochastic differential equations generating 1/f(β) noise is established and analyzed, as well.
Donnet, Sophie; Samson, Adeline
2013-06-30
This paper is a survey of existing estimation methods for pharmacokinetic/pharmacodynamic (PK/PD) models based on stochastic differential equations (SDEs). Most parametric estimation methods proposed for SDEs require high frequency data and are often poorly suited for PK/PD data which are usually sparse. Moreover, PK/PD experiments generally include not a single individual but a group of subjects, leading to a population estimation approach. This review concentrates on estimation methods which have been applied to PK/PD data, for SDEs observed with and without measurement noise, with a standard or a population approach. Besides, the adopted methodologies highly differ depending on the existence or not of an explicit transition density of the SDE solution. Copyright © 2013 Elsevier B.V. All rights reserved.
Stochastic Navier-Stokes Equations in Unbounded Channel Domains (Open Source)
2014-09-17
inequality (6.10) is a martingale with zero expectation, one obtains E [|vn(t ∧ τN )|2e−δt ] + νE (∫ t∧τN 0 ‖vn(s)‖2e−δs ds ) ≤ 2E { E (∫ t∧τN 0...t)Tr(g∗g)(t) dt ] . But 2 ∫ T 0 e−r(t)(g(t) dW (t),vn(t)) is a martingale having expectation zero. Then from the above equation, we have E [ e−r(T...Lévy noise. Stoch. Anal. Appl. 31, 1–46 (2013) [24] Flandoli, F., Gatarek, D.: Martingale and stationary solutions for stochastic Navier–Stokes
A stochastic differential equation model for the foraging behavior of fish schools
Tạ, Tôn ệt, Vi; Hoai Nguyen, Linh Thi
2018-05-01
Constructing models of living organisms locating food sources has important implications for understanding animal behavior and for the development of distribution technologies. This paper presents a novel simple model of stochastic differential equations for the foraging behavior of fish schools in a space including obstacles. The model is studied numerically. Three configurations of space with various food locations are considered. In the first configuration, fish swim in free but limited space. All individuals can find food with large probability while keeping their school structure. In the second and third configurations, they move in limited space with one and two obstacles, respectively. Our results reveal that the probability of foraging success is highest in the first configuration, and smallest in the third one. Furthermore, when school size increases up to an optimal value, the probability of foraging success tends to increase. When it exceeds an optimal value, the probability tends to decrease. The results agree with experimental observations.
Kinetics of subdiffusion-assisted reactions: non-Markovian stochastic Liouville equation approach
Energy Technology Data Exchange (ETDEWEB)
Shushin, A I [Institute of Chemical Physics, Russian Academy of Sciences, 117977, GSP-1, Kosygin str. 4, Moscow (Russian Federation)
2005-01-01
Anomalous specific features of the kinetics of subdiffusion-assisted bimolecular reactions (time-dependence, dependence on parameters of systems, etc) are analysed in detail with the use of the non-Markovian stochastic Liouville equation (SLE), which has been recently derived within the continuous-time random-walk (CTRW) approach. In the CTRW approach, subdiffusive motion of particles is modelled by jumps whose onset probability distribution function is of a long-tailed form. The non-Markovian SLE allows for rigorous describing of some peculiarities of these reactions; for example, very slow long-time behaviour of the kinetics, non-analytical dependence of the reaction rate on the reactivity of particles, strong manifestation of fluctuation kinetics showing itself in very slowly decreasing behaviour of the kinetics at very long times, etc.
Directory of Open Access Journals (Sweden)
Soloviev Igor A.
2016-01-01
Full Text Available Stochastic differential equations are proposed on the basis of generalized Fokker-Planck-Kolmogorov equation. From the statements of boundary and initial value problems for probabilistic density, dispersion and differential entropy the conditions for stability of solutions and changes in scenarios of development of random phenomena of heat and nuclear particle transfer are obtained. [Projekat Ministartsva nauke Republike Srbije, br. 174005 i br. 174024
Directory of Open Access Journals (Sweden)
2006-01-01
Full Text Available We derive necessary and sufficient conditions for global-in-time existence of solutions of ordinary differential, stochastic differential, and parabolic equations. The conditions are formulated in terms of complete Riemannian metrics on extended phase spaces (conditions with two-sided estimates or in terms of derivatives of proper functions on extended phase spaces (conditions with one-sided estimates.
DEFF Research Database (Denmark)
Møller, Jonas Bech; Overgaard, R.V.; Madsen, Henrik
2010-01-01
Several articles have investigated stochastic differential equations (SDEs) in PK/PD models, but few have quantitatively investigated the benefits to predictive performance of models based on real data. Estimation of first phase insulin secretion which reflects beta-cell function using models...
Overgaard, Rune V; Jonsson, Niclas; Tornøe, Christoffer W; Madsen, Henrik
2005-02-01
Pharmacokinetic/pharmacodynamic modelling is most often performed using non-linear mixed-effects models based on ordinary differential equations with uncorrelated intra-individual residuals. More sophisticated residual error models as e.g. stochastic differential equations (SDEs) with measurement noise can in many cases provide a better description of the variations, which could be useful in various aspects of modelling. This general approach enables a decomposition of the intra-individual residual variation epsilon into system noise w and measurement noise e. The present work describes implementation of SDEs in a non-linear mixed-effects model, where parameter estimation was performed by a novel approximation of the likelihood function. This approximation is constructed by combining the First-Order Conditional Estimation (FOCE) method used in non-linear mixed-effects modelling with the Extended Kalman Filter used in models with SDEs. Fundamental issues concerning the proposed model and estimation algorithm are addressed by simulation studies, concluding that system noise can successfully be separated from measurement noise and inter-individual variability.
Lipan, Ovidiu; Ferwerda, Cameron
2018-02-01
The deterministic Hill function depends only on the average values of molecule numbers. To account for the fluctuations in the molecule numbers, the argument of the Hill function needs to contain the means, the standard deviations, and the correlations. Here we present a method that allows for stochastic Hill functions to be constructed from the dynamical evolution of stochastic biocircuits with specific topologies. These stochastic Hill functions are presented in a closed analytical form so that they can be easily incorporated in models for large genetic regulatory networks. Using a repressive biocircuit as an example, we show by Monte Carlo simulations that the traditional deterministic Hill function inaccurately predicts time of repression by an order of two magnitudes. However, the stochastic Hill function was able to capture the fluctuations and thus accurately predicted the time of repression.
Directory of Open Access Journals (Sweden)
O. V. Borisenko
1994-01-01
Full Text Available Using connection between stochastic differential equation with Poisson measure term and its Kolmogorov's equation, we investigate the limiting behavior of the Cauchy problem solution of the integro differential equation with coefficients depending on a small parameter. We also study the dependence of the limiting equation on the order of the parameter.
International Nuclear Information System (INIS)
Girko, V.L.
2011-01-01
Our studies are essentially based on the martingale differences method developed in my previous papers for resolvents of random matrices. This method possesses the self-averaging property of the entries of resolvents of random matrices and, hence, we can deduce the stochastic canonical equation. The lecture contains the most important results from numerous papers and books dealing with the theory of Unitary random matrices and functions of random matrices. We give the REFORM method of proving of all results, avoiding the method of moments. We do not try to describe here all known properties of the eigenvalues and eigenvectors for all classes of random matrices. However, our aim is rather to present the theory of stochastic canonical equations, and to give rigorous proofs of the procedures used to deduce these equations on the base of the author's General Statistical Analysis. We consider special classes of analytic functions of random matrices. The description problem for normalized spectral functions of some analytic functions of random matrices is discussed in detail. Specifically, we present here the new theory: LIFE, which is the abbreviation for Limit Independence of Functions of Ensembles. (author)
Averaging of Differential Equations Generating Oscillations and an Application to Control
International Nuclear Information System (INIS)
Temam, R.M.; Wirosoetisno, D.
2002-01-01
In this article we consider differential equations which generate oscillating solutions. These oscillations are due to the presence of a small parameter ε>0 ; however, they are not present in the coefficients but instead they are caused by a penalty term involving an antisymmetric operator. Our aims are twofold. In the first part we study asymptotics at all orders, for ε → 0 , construct approximate solutions, and derive estimates of the error between the exact solution and the approximate ones. One of the motivations of this part is the study to high orders of the geostrophic asymptotics in atmospheric science, but there are many other possible applications involving in particular the wave equation. The actual applications of our results to atmospheric science will be discussed elsewhere [STW], as well as, on the mathematical side, the application to partial differential equations [TW1]. In the second part of this article we study a control problem involving such an equation and study the behavior of the state equation, of the optimal control, and of the optimality equation as ε → 0 . For the control part we restrict ourselves to a linear equation and to the first order in the asymptotics ε → 0 , leaving nonlinear problems and higher orders to a future work
Research on nonlinear stochastic dynamical price model
International Nuclear Information System (INIS)
Li Jiaorui; Xu Wei; Xie Wenxian; Ren Zhengzheng
2008-01-01
In consideration of many uncertain factors existing in economic system, nonlinear stochastic dynamical price model which is subjected to Gaussian white noise excitation is proposed based on deterministic model. One-dimensional averaged Ito stochastic differential equation for the model is derived by using the stochastic averaging method, and applied to investigate the stability of the trivial solution and the first-passage failure of the stochastic price model. The stochastic price model and the methods presented in this paper are verified by numerical studies
International Nuclear Information System (INIS)
Ray A. Berry
2005-01-01
At the INL researchers and engineers routinely encounter multiphase, multi-component, and/or multi-material flows. Some examples include: Reactor coolant flows Molten corium flows Dynamic compaction of metal powders Spray forming and thermal plasma spraying Plasma quench reactor Subsurface flows, particularly in the vadose zone Internal flows within fuel cells Black liquor atomization and combustion Wheat-chaff classification in combine harvesters Generation IV pebble bed, high temperature gas reactor The complexity of these flows dictates that they be examined in an averaged sense. Typically one would begin with known (or at least postulated) microscopic flow relations that hold on the ''small'' scale. These include continuum level conservation of mass, balance of species mass and momentum, conservation of energy, and a statement of the second law of thermodynamics often in the form of an entropy inequality (such as the Clausius-Duhem inequality). The averaged or macroscopic conservation equations and entropy inequalities are then obtained from the microscopic equations through suitable averaging procedures. At this stage a stronger form of the second law may also be postulated for the mixture of phases or materials. To render the evolutionary material flow balance system unique, constitutive equations and phase or material interaction relations are introduced from experimental observation, or by postulation, through strict enforcement of the constraints or restrictions resulting from the averaged entropy inequalities. These averaged equations form the governing equation system for the dynamic evolution of these mixture flows. Most commonly, the averaging technique utilized is either volume or time averaging or a combination of the two. The flow restrictions required for volume and time averaging to be valid can be severe, and violations of these restrictions are often found. A more general, less restrictive (and far less commonly used) type of averaging known as
Interpretation of non-Markovian stochastic Schroedinger equations as a hidden-variable theory
International Nuclear Information System (INIS)
Gambetta, Jay; Wiseman, H.M.
2003-01-01
Do diffusive non-Markovian stochastic Schroedinger equations (SSEs) for open quantum systems have a physical interpretation? In a recent paper [Phys. Rev. A 66, 012108 (2002)] we investigated this question using the orthodox interpretation of quantum mechanics. We found that the solution of a non-Markovian SSE represents the state the system would be in at that time if a measurement was performed on the environment at that time, and yielded a particular result. However, the linking of solutions at different times to make a trajectory is, we concluded, a fiction. In this paper we investigate this question using the modal (hidden variable) interpretation of quantum mechanics. We find that the noise function z(t) appearing in the non-Markovian SSE can be interpreted as a hidden variable for the environment. That is, some chosen property (beable) of the environment has a definite value z(t) even in the absence of measurement on the environment. The non-Markovian SSE gives the evolution of the state of the system 'conditioned' on this environment hidden variable. We present the theory for diffusive non-Markovian SSEs that have as their Markovian limit SSEs corresponding to homodyne and heterodyne detection, as well as one which has no Markovian limit
THE GROSSER ALETSCHGLETSCHER DYNAMICS: FROM A “MINIMAL MODEL” TO A STOCHASTIC EQUATION
Directory of Open Access Journals (Sweden)
Alexander V. Kislov
2016-01-01
Full Text Available Mountain glaciers manifest oscillations at different time-scales. Apart from synchronous reaction to lasting changes, there is asynchronism between climatic forcing and observed anomalies of the glaciers. Based on general theories on the laws of temporal dynamics relating to massive inertial objects, the observed interannual changes of glacier length could result from the accumulation of small anomalies in the heat/water fluxes. Despite the fact that the original model of the dynamics of mountain glaciers is deterministically based on the physical law of conservation of water mass, the model of length change is interpreted as stochastic; from this perspective, it is the Langevin equation that incorporates the action of temperature anomalies and precipitation like random white noise. The process is analogous to Brownian motion. Under these conditions, the Grosser Aletschgletscher (selected as an example is represented by a system undergoing a random walk. It was shown that the possible range of variability covers the observed interval of length fluctuations.
Møller, Jan Kloppenborg; Bergmann, Kirsten Riber; Christiansen, Lasse Engbo; Madsen, Henrik
2012-07-21
In the present study, bacterial growth in a rich media is analysed in a Stochastic Differential Equation (SDE) framework. It is demonstrated that the SDE formulation and smoothened state estimates provide a systematic framework for data driven model improvements, using random walk hidden states. Bacterial growth is limited by the available substrate and the inclusion of diffusion must obey this natural restriction. By inclusion of a modified logistic diffusion term it is possible to introduce a diffusion term flexible enough to capture both the growth phase and the stationary phase, while concentration is restricted to the natural state space (substrate and bacteria non-negative). The case considered is the growth of Salmonella and Enterococcus in a rich media. It is found that a hidden state is necessary to capture the lag phase of growth, and that a flexible logistic diffusion term is needed to capture the random behaviour of the growth model. Further, it is concluded that the Monod effect is not needed to capture the dynamics of bacterial growth in the data presented. Copyright © 2012 Elsevier Ltd. All rights reserved.
A stochastic collocation method for the second order wave equation with a discontinuous random speed
Motamed, Mohammad
2012-08-31
In this paper we propose and analyze a stochastic collocation method for solving the second order wave equation with a random wave speed and subjected to deterministic boundary and initial conditions. The speed is piecewise smooth in the physical space and depends on a finite number of random variables. The numerical scheme consists of a finite difference or finite element method in the physical space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space. This approach leads to the solution of uncoupled deterministic problems as in the Monte Carlo method. We consider both full and sparse tensor product spaces of orthogonal polynomials. We provide a rigorous convergence analysis and demonstrate different types of convergence of the probability error with respect to the number of collocation points for full and sparse tensor product spaces and under some regularity assumptions on the data. In particular, we show that, unlike in elliptic and parabolic problems, the solution to hyperbolic problems is not in general analytic with respect to the random variables. Therefore, the rate of convergence may only be algebraic. An exponential/fast rate of convergence is still possible for some quantities of interest and for the wave solution with particular types of data. We present numerical examples, which confirm the analysis and show that the collocation method is a valid alternative to the more traditional Monte Carlo method for this class of problems. © 2012 Springer-Verlag.
Stochastic differential equation (SDE) model of opening gold share price of bursa saham malaysia
Hussin, F. N.; Rahman, H. A.; Bahar, A.
2017-09-01
Black and Scholes option pricing model is one of the most recognized stochastic differential equation model in mathematical finance. Two parameter estimation methods have been utilized for the Geometric Brownian model (GBM); historical and discrete method. The historical method is a statistical method which uses the property of independence and normality logarithmic return, giving out the simplest parameter estimation. Meanwhile, discrete method considers the function of density of transition from the process of diffusion normal log which has been derived from maximum likelihood method. These two methods are used to find the parameter estimates samples of Malaysians Gold Share Price data such as: Financial Times and Stock Exchange (FTSE) Bursa Malaysia Emas, and Financial Times and Stock Exchange (FTSE) Bursa Malaysia Emas Shariah. Modelling of gold share price is essential since fluctuation of gold affects worldwide economy nowadays, including Malaysia. It is found that discrete method gives the best parameter estimates than historical method due to the smallest Root Mean Square Error (RMSE) value.
Asymptotically Almost Periodic Solutions for a Class of Stochastic Functional Differential Equations
Directory of Open Access Journals (Sweden)
Aimin Liu
2014-01-01
Full Text Available This work is concerned with the quadratic-mean asymptotically almost periodic mild solutions for a class of stochastic functional differential equations dxt=Atxt+Ft,xt,xtdt+H(t,xt,xt∘dW(t. A new criterion ensuring the existence and uniqueness of the quadratic-mean asymptotically almost periodic mild solutions for the system is presented. The condition of being uniformly exponentially stable of the strongly continuous semigroup {Tt}t≥0 is essentially removed, which is generated by the linear densely defined operator A∶D(A⊂L2(ℙ,ℍ→L2(ℙ,ℍ, only using the exponential trichotomy of the system, which reflects a deeper analysis of the behavior of solutions of the system. In this case the asymptotic behavior is described through the splitting of the main space into stable, unstable, and central subspaces at each point from the flow’s domain. An example is also given to illustrate our results.
Bayesian analysis of growth curves using mixed models defined by stochastic differential equations.
Donnet, Sophie; Foulley, Jean-Louis; Samson, Adeline
2010-09-01
Growth curve data consist of repeated measurements of a continuous growth process over time in a population of individuals. These data are classically analyzed by nonlinear mixed models. However, the standard growth functions used in this context prescribe monotone increasing growth and can fail to model unexpected changes in growth rates. We propose to model these variations using stochastic differential equations (SDEs) that are deduced from the standard deterministic growth function by adding random variations to the growth dynamics. A Bayesian inference of the parameters of these SDE mixed models is developed. In the case when the SDE has an explicit solution, we describe an easily implemented Gibbs algorithm. When the conditional distribution of the diffusion process has no explicit form, we propose to approximate it using the Euler-Maruyama scheme. Finally, we suggest validating the SDE approach via criteria based on the predictive posterior distribution. We illustrate the efficiency of our method using the Gompertz function to model data on chicken growth, the modeling being improved by the SDE approach. © 2009 INRA, Government of France.
Stochastic differential equations for evolutionary dynamics with demographic noise and mutations.
Traulsen, Arne; Claussen, Jens Christian; Hauert, Christoph
2012-04-01
We present a general framework to describe the evolutionary dynamics of an arbitrary number of types in finite populations based on stochastic differential equations (SDEs). For large, but finite populations this allows us to include demographic noise without requiring explicit simulations. Instead, the population size only rescales the amplitude of the noise. Moreover, this framework admits the inclusion of mutations between different types, provided that mutation rates μ are not too small compared to the inverse population size 1/N. This ensures that all types are almost always represented in the population and that the occasional extinction of one type does not result in an extended absence of that type. For μN≪1 this limits the use of SDEs, but in this case there are well established alternative approximations based on time scale separation. We illustrate our approach by a rock-scissors-paper game with mutations, where we demonstrate excellent agreement with simulation based results for sufficiently large populations. In the absence of mutations the excellent agreement extends to small population sizes.
International Nuclear Information System (INIS)
Hatlee, M.D.; Kozak, J.J.
1981-01-01
In this paper we focus on a particular class of intramicellar kinetic processes: reversible reactions of molecularity two taking place in the interior (or perhaps the immediate vicinity) of a micellar assembly. We formulate a stochastic master equation to describe an ensemble of compartmentalized, distributed systems wherein the (only) microscopic chemical event is a photoinduced, reversible, bimolecular reaction. Then, for an initial distribution of reactants assumed to be Poissonian, we calculate and characterize the overall (macroscopic) dynamics displayed by such a system. The resulting kinetic description is compared with the behavior found previously for the simpler case of irreversible, intramicellar kinetic processes. Finally, we introduce and then investigate the notion of an ''apparent'' equilibrium constant Q for a reversible reaction taking place in a compartmentalized, distributed system and compare this quantity with the ''canonical'' equilibrium constant K obtained for the (same) reversible reaction carried out in bulk, homogeneous solution. The full behavior of Q=Q(K) is explored numerically, and we prove analytically that Qapprox.K 2 in the small K limit and that Q is independent of K in the limit of large K. The experimental consequences of the theory are dicussed in some detail
Flohr, J R; Dritz, S S; Tokach, M D; Woodworth, J C; DeRouchey, J M; Goodband, R D
2018-05-01
Floor space allowance for pigs has substantial effects on pig growth and welfare. Data from 30 papers examining the influence of floor space allowance on the growth of finishing pigs was used in a meta-analysis to develop alternative prediction equations for average daily gain (ADG), average daily feed intake (ADFI) and gain : feed ratio (G : F). Treatment means were compiled in a database that contained 30 papers for ADG and 28 papers for ADFI and G : F. The predictor variables evaluated were floor space (m2/pig), k (floor space/final BW0.67), Initial BW, Final BW, feed space (pigs per feeder hole), water space (pigs per waterer), group size (pigs per pen), gender, floor type and study length (d). Multivariable general linear mixed model regression equations were used. Floor space treatments within each experiment were the observational and experimental unit. The optimum equations to predict ADG, ADFI and G : F were: ADG, g=337.57+(16 468×k)-(237 350×k 2)-(3.1209×initial BW (kg))+(2.569×final BW (kg))+(71.6918×k×initial BW (kg)); ADFI, g=833.41+(24 785×k)-(388 998×k 2)-(3.0027×initial BW (kg))+(11.246×final BW (kg))+(187.61×k×initial BW (kg)); G : F=predicted ADG/predicted ADFI. Overall, the meta-analysis indicates that BW is an important predictor of ADG and ADFI even after computing the constant coefficient k, which utilizes final BW in its calculation. This suggests including initial and final BW improves the prediction over using k as a predictor alone. In addition, the analysis also indicated that G : F of finishing pigs is influenced by floor space allowance, whereas individual studies have concluded variable results.
Energy Technology Data Exchange (ETDEWEB)
Benseghir, Rym, E-mail: benseghirrym@ymail.com, E-mail: benseghirrym@ymail.com; Benchettah, Azzedine, E-mail: abenchettah@hotmail.com [LANOS Laboratory, Badji Mokhtar University, BP 12, 23000, Annaba (Algeria); Raynaud de Fitte, Paul, E-mail: prf@univ-rouen.fr [Normandie Univ, Laboratoire Raphaël Salem, UMR CNRS 6085, Rouen (France)
2015-11-30
A stochastic equation system corresponding to the description of the motion of a barotropic viscous gas in a discretized one-dimensional domain with a weight regularizing the density is considered. In [2], the existence of an invariant measure was established for this discretized problem in the stationary case. In this paper, applying a slightly modified version of Khas’minskii’s theorem [5], we generalize this result in the periodic case by proving the existence of a periodic measure for this problem.
Crisan, Dan
2011-01-01
"Stochastic Analysis" aims to provide mathematical tools to describe and model high dimensional random systems. Such tools arise in the study of Stochastic Differential Equations and Stochastic Partial Differential Equations, Infinite Dimensional Stochastic Geometry, Random Media and Interacting Particle Systems, Super-processes, Stochastic Filtering, Mathematical Finance, etc. Stochastic Analysis has emerged as a core area of late 20th century Mathematics and is currently undergoing a rapid scientific development. The special volume "Stochastic Analysis 2010" provides a sa
Energy Technology Data Exchange (ETDEWEB)
Suescun D, D.; Oviedo T, M., E-mail: daniel.suescun@usco.edu.co [Universidad Surcolombiana, Av. Pastrana Borrero - Carrera 1, Neiva, Huila (Colombia)
2017-09-15
In this paper, a numerical study of stochastic differential equations that describe the kinetics in a nuclear reactor is presented. These equations, known as the stochastic equations of punctual kinetics they model temporal variations in neutron population density and concentrations of deferred neutron precursors. Because these equations are probabilistic in nature (since random oscillations in the neutrons and population of precursors were considered to be approximately normally distributed, and these equations also possess strong coupling and stiffness properties) the proposed method for the numerical simulations is the Euler-Maruyama scheme that provides very good approximations for calculating the neutron population and concentrations of deferred neutron precursors. The method proposed for this work was computationally tested for different seeds, initial conditions, experimental data and forms of reactivity for a group of precursors and then for six groups of deferred neutron precursors at each time step with 5000 Brownian movements per seed. In a paper reported in the literature, the Euler-Maruyama method was proposed, but there are many doubts about the reported values, in addition to not reporting the seed used, so in this work is expected to rectify the reported values. After taking the average of the different seeds used to generate the pseudo-random numbers the results provided by the Euler-Maruyama scheme will be compared in mean and standard deviation with other methods reported in the literature and results of the deterministic model of the equations of the punctual kinetics. This comparison confirms in particular that the Euler-Maruyama scheme is an efficient method to solve the equations of stochastic point kinetics but different from the values found and reported by another author. The Euler-Maruyama method is simple and easy to implement, provides acceptable results for neutron population density and concentration of deferred neutron precursors and
DEFF Research Database (Denmark)
Overgaard, Rune Viig; Holford, Nick; Rytved, K. A.
2007-01-01
Purpose To describe the pharmacodynamic effects of recombinant human interleukin-21 (IL-21) on core body temperature in cynomolgus monkeys using basic mechanisms of heat regulation. A major effort was devoted to compare the use of ordinary differential equations (ODEs) with stochastic differential...... equations (SDEs) in pharmacokinetic pharmacodynamic (PKPD) modelling. Methods A temperature model was formulated including circadian rhythm, metabolism, heat loss, and a thermoregulatory set-point. This model was formulated as a mixed-effects model based on SDEs using NONMEM. Results The effects of IL-21...
Wood, Brian; He, Xiaoliang; Apte, Sourabh
2017-11-01
Turbulent flows through porous media are encountered in a number of natural and engineered systems. Many attempts to close the Navier-Stokes equation for such type of flow have been made, for example using RANS models and double averaging. On the other hand, Whitaker (1996) applied volume averaging theorem to close the macroscopic N-S equation for low Re flow. In this work, the volume averaging theory is extended into the turbulent flow regime to posit a relationship between the macroscale velocities and the spatial velocity statistics in terms of the spatial averaged velocity only. Rather than developing a Reynolds stress model, we propose a simple algebraic closure, consistent with generalized effective viscosity models (Pope 1975), to represent the spatial fluctuating velocity and pressure respectively. The coefficients (one 1st order, two 2nd order and one 3rd order tensor) of the linear functions depend on averaged velocity and gradient. With the data set from DNS, performed with inertial and turbulent flows (pore Re of 300, 500 and 1000) through a periodic face centered cubic (FCC) unit cell, all the unknown coefficients can be computed and the closure is complete. The macroscopic quantity calculated from the averaging is then compared with DNS data to verify the upscaling. NSF Project Numbers 1336983, 1133363.
International Nuclear Information System (INIS)
Laurence, D.
1997-01-01
The k-ε model and Reynolds stress transport model are set out in a few words. Limitations of models are shown, particularly for turbulence generation in the turbulent viscosity context, and, more generally, the uncertainties and miscellaneous changes made to the dissipation equation. The performances of models are then compared, using results of the three latest ERCOFTA/IAHR workshops. It is shown that algebraic constraints which can be derived exactly by assuming asymptotic limits (rapid distortion, homogeneous shear at infinite time, 2D turbulence) have inhibited a better tuning of the models for real life flow where these limits are not encountered. A more pragmatic approach could be taken by allowing the constants to be functions of invariant parameters. But these functions, making the models non-linear, can lead to bifurcations or instability. One essential parameter is the distance to the wall, which recent models have tried to eliminate, although this parameter appears indirectly through the Poisson equation for the fluctuating pressure. A possible indirect model is the elliptic relaxation. Progress was recently achieved in near-wall low Re modelling, but these advances do not always result in benefits to industry since only the 'wall function' approaches can be used in the high Re, 3D flows that we need to study. With the knowledge gained from near-wall modelling, it might be profitable to revisit the 'wall functions' devised 20 years ago. (author)
Cosso, Andrea; Russo, Francesco
2016-11-01
Functional Itô calculus was introduced in order to expand a functional F(t,Xṡ+t,Xt) depending on time t, past and present values of the process X. Another possibility to expand F(t,Xṡ+t,Xt) consists in considering the path Xṡ+t = {Xx+t,x ∈ [-T, 0]} as an element of the Banach space of continuous functions on C([-T, 0]) and to use Banach space stochastic calculus. The aim of this paper is threefold. (1) To reformulate functional Itô calculus, separating time and past, making use of the regularization procedures which match more naturally the notion of horizontal derivative which is one of the tools of that calculus. (2) To exploit this reformulation in order to discuss the (not obvious) relation between the functional and the Banach space approaches. (3) To study existence and uniqueness of smooth solutions to path-dependent partial differential equations which naturally arise in the study of functional Itô calculus. More precisely, we study a path-dependent equation of Kolmogorov type which is related to the window process of the solution to an Itô stochastic differential equation with path-dependent coefficients. We also study a semilinear version of that equation.
Deng, Chenhui; Plan, Elodie L; Karlsson, Mats O
2016-06-01
Parameter variation in pharmacometric analysis studies can be characterized as within subject parameter variability (WSV) in pharmacometric models. WSV has previously been successfully modeled using inter-occasion variability (IOV), but also stochastic differential equations (SDEs). In this study, two approaches, dynamic inter-occasion variability (dIOV) and adapted stochastic differential equations, were proposed to investigate WSV in pharmacometric count data analysis. These approaches were applied to published count models for seizure counts and Likert pain scores. Both approaches improved the model fits significantly. In addition, stochastic simulation and estimation were used to explore further the capability of the two approaches to diagnose and improve models where existing WSV is not recognized. The results of simulations confirmed the gain in introducing WSV as dIOV and SDEs when parameters vary randomly over time. Further, the approaches were also informative as diagnostics of model misspecification, when parameters changed systematically over time but this was not recognized in the structural model. The proposed approaches in this study offer strategies to characterize WSV and are not restricted to count data.
Average value of the shape and direction factor in the equation of refractive index
Zhang, Tao
2017-10-01
The theoretical calculation of the refractive indices is of great significance for the developments of new optical materials. The calculation method of refractive index, which was deduced from the electron-cloud-conductor model, contains the shape and direction factor 〈g〉. 〈g〉 affects the electromagnetic-induction energy absorbed by the electron clouds, thereby influencing the refractive indices. It is not yet known how to calculate 〈g〉 value of non-spherical electron clouds. In this paper, 〈g〉 value is derived by imaginatively dividing the electron cloud into numerous little volume elements and then regrouping them. This paper proves that 〈g〉 = 2/3 when molecules’ spatial orientations distribute randomly. The calculations of the refractive indices of several substances validate this equation. This result will help to promote the application of the calculation method of refractive index.
Langevin equation with time dependent linear force and periodic load force: stochastic resonance
Sau Fa, Kwok
2017-11-01
The motion of a particle described by the Langevin equation with constant diffusion coefficient, time dependent linear force (ω (1+α \\cos ({ω }1t))x) and periodic load force ({A}0\\cos ({{Ω }}t)) is investigated. Analytical solutions for the probability density function (PDF) and n-moment are obtained and analysed. For {ω }1\\gg α ω the influence of the periodic term α \\cos ({ω }1t) is negligible to the PDF and n-moment for any time; this result shows that the statistical averages such as n-moments and the PDF have no access to some information of the system. For small and intermediate values of {ω }1 the influence of the periodic term α \\cos ({ω }1t) to the system is also analysed; in particular the system may present multiresonance. The solutions are obtained in a direct and pedagogical manner readily understandable by graduate students.
Energy Technology Data Exchange (ETDEWEB)
Webster, Clayton; Tempone, Raul (Florida State University, Tallahassee, FL); Nobile, Fabio (Politecnico di Milano, Italy)
2007-12-01
This work describes the convergence analysis of a Smolyak-type sparse grid stochastic collocation method for the approximation of statistical quantities related to the solution of partial differential equations with random coefficients and forcing terms (input data of the model). To compute solution statistics, the sparse grid stochastic collocation method uses approximate solutions, produced here by finite elements, corresponding to a deterministic set of points in the random input space. This naturally requires solving uncoupled deterministic problems and, as such, the derived strong error estimates for the fully discrete solution are used to compare the computational efficiency of the proposed method with the Monte Carlo method. Numerical examples illustrate the theoretical results and are used to compare this approach with several others, including the standard Monte Carlo.
Rouz, Omid Farkhondeh; Ahmadian, Davood; Milev, Mariyan
2017-12-01
This paper establishes exponential mean square stability of two classes of theta Milstein methods, namely split-step theta Milstein (SSTM) method and stochastic theta Milstein (STM) method, for stochastic differential delay equations (SDDEs). We consider the SDDEs problem under a coupled monotone condition on drift and diffusion coefficients, as well as a necessary linear growth condition on the last term of theta Milstein method. It is proved that the SSTM method with θ ∈ [0, ½] can recover the exponential mean square stability of the exact solution with some restrictive conditions on stepsize, but for θ ∈ (½, 1], we proved that the stability results hold for any stepsize. Then, based on the stability results of SSTM method, we examine the exponential mean square stability of the STM method and obtain the similar stability results to that of the SSTM method. In the numerical section the figures show thevalidity of our claims.
Ghosh, Arup Kumar; Hussain, Faraz; Jha, Susmit; Langmead, Christopher J; Jha, Sumit Kumar
2014-01-01
Stochastic Differential Equation (SDE) models are used to describe the dynamics of complex systems with inherent randomness. The primary purpose of these models is to study rare but interesting or important behaviours, such as the formation of a tumour. Stochastic simulations are the most common means for estimating (or bounding) the probability of rare behaviours, but the cost of simulations increases with the rarity of events. To address this problem, we introduce a new algorithm specifically designed to quantify the likelihood of rare behaviours in SDE models. Our approach relies on temporal logics for specifying rare behaviours of interest, and on the ability of bit-vector decision procedures to reason exhaustively about fixed-precision arithmetic. We apply our algorithm to a minimal parameterised model of the cell cycle, and take Brownian noise into account while investigating the likelihood of irregularities in cell size and time between cell divisions.
Preston, L. A.
2017-12-01
Marine hydrokinetic (MHK) devices offer a clean, renewable alternative energy source for the future. Responsible utilization of MHK devices, however, requires that the effects of acoustic noise produced by these devices on marine life and marine-related human activities be well understood. Paracousti is a 3-D full waveform acoustic modeling suite that can accurately propagate MHK noise signals in the complex bathymetry found in the near-shore to open ocean environment and considers real properties of the seabed, water column, and air-surface interface. However, this is a deterministic simulation that assumes the environment and source are exactly known. In reality, environmental and source characteristics are often only known in a statistical sense. Thus, to fully characterize the expected noise levels within the marine environment, this uncertainty in environmental and source factors should be incorporated into the acoustic simulations. One method is to use Monte Carlo (MC) techniques where simulation results from a large number of deterministic solutions are aggregated to provide statistical properties of the output signal. However, MC methods can be computationally prohibitive since they can require tens of thousands or more simulations to build up an accurate representation of those statistical properties. An alternative method, using the technique of stochastic partial differential equations (SPDE), allows computation of the statistical properties of output signals at a small fraction of the computational cost of MC. We are developing a SPDE solver for the 3-D acoustic wave propagation problem called Paracousti-UQ to help regulators and operators assess the statistical properties of environmental noise produced by MHK devices. In this presentation, we present the SPDE method and compare statistical distributions of simulated acoustic signals in simple models to MC simulations to show the accuracy and efficiency of the SPDE method. Sandia National Laboratories
Comparative study of dense plasma state equations obtained from different models of average-atom
International Nuclear Information System (INIS)
Fromy, Patrice
1991-01-01
This research thesis addresses the influence of temperature and density effects on magnitudes such as pressure, energy, ionisation, and on energy levels of a body described according to the approximation of an electrically neutral isolated atomic sphere. Starting from the general formalism of the functional density, with some approximations, the author deduces the Thomas-Fermi, Thomas-Fermi-Dirac, and Thomas-Fermi-Dirac-Weizsaecker models, and an average-atom approximated quantum model. For each of these models, the author presents an explicit method of resolution, as well as the determination of different magnitudes taken into account in this study. For the different studied magnitudes, the author highlights effects due to the influence of temperature and of density, as well as variations due to the different models [fr
Sotiropoulos, Vassilios; Kaznessis, Yiannis N
2008-01-07
Models involving stochastic differential equations (SDEs) play a prominent role in a wide range of applications where systems are not at the thermodynamic limit, for example, biological population dynamics. Therefore there is a need for numerical schemes that are capable of accurately and efficiently integrating systems of SDEs. In this work we introduce a variable size step algorithm and apply it to systems of stiff SDEs with multiple multiplicative noise. The algorithm is validated using a subclass of SDEs called chemical Langevin equations that appear in the description of dilute chemical kinetics models, with important applications mainly in biology. Three representative examples are used to test and report on the behavior of the proposed scheme. We demonstrate the advantages and disadvantages over fixed time step integration schemes of the proposed method, showing that the adaptive time step method is considerably more stable than fixed step methods with no excessive additional computational overhead.
Dallas, S. S.; Khan, I.
1976-01-01
The singly-averaged differential equations of motion of a satellite are developed in terms of parameters valid for all eccentricities less than one. The perturbations included in the acceleration model are due to an aspherical central planet (zonal harmonics up to degree 20 and resonant harmonics up to degree and order 20), atmospheric drag for a time-varying atmosphere, third-body gravity (the sun and moon for an earth satellite), solar radiation pressure with shadowing, and impulsive maneuvers. Analytic averaging is used to remove short-period terms due to the aspherical central planet and third-body gravity. Numerical averaging is used to remove short-period terms due to atmospheric drag and solar radiation pressure.
Caglar, Mehmet Umut; Pal, Ranadip
2011-03-01
Central dogma of molecular biology states that ``information cannot be transferred back from protein to either protein or nucleic acid''. However, this assumption is not exactly correct in most of the cases. There are a lot of feedback loops and interactions between different levels of systems. These types of interactions are hard to analyze due to the lack of cell level data and probabilistic - nonlinear nature of interactions. Several models widely used to analyze and simulate these types of nonlinear interactions. Stochastic Master Equation (SME) models give probabilistic nature of the interactions in a detailed manner, with a high calculation cost. On the other hand Probabilistic Boolean Network (PBN) models give a coarse scale picture of the stochastic processes, with a less calculation cost. Differential Equation (DE) models give the time evolution of mean values of processes in a highly cost effective way. The understanding of the relations between the predictions of these models is important to understand the reliability of the simulations of genetic regulatory networks. In this work the success of the mapping between SME, PBN and DE models is analyzed and the accuracy and affectivity of the control policies generated by using PBN and DE models is compared.
Energy Technology Data Exchange (ETDEWEB)
Davidson, R.C.; Lee, W.W.; Stoltz, P.
1997-08-01
This paper presents a detailed formulation and analysis of the rate equations for statistically-averaged quantities for an intense nonneutral beam propagating through a periodic solenoidal focusing field B{sup sol}(x). The analysis is based on the nonlinear Vlasov-Maxwell equations in the electrostatic approximation, assuming a thin beam with characteristic beam radius r{sub b} {much_lt} S. The results are applied to investigate the nonlinear evolution of the generalized entropy, mean canonical angular momentum {l_angle}P{sub {theta}}{r_angle}, center-of-mass motion for {l_angle}X{r_angle} and {l_angle}Y{r_angle}, mean kinetic energy (1/2) {l_angle}X{sup {prime}2} + Y{sup {prime}2}{r_angle}, mean-square beam radius {l_angle}X{sup 2} + Y{sup 2}{r_angle}, and coupled rate equations for the unnormalized transverse emittance {epsilon}(s) and root-mean-square beam radius R{sub b}(s) = {l_angle}X{sup 2} + Y{sup 2}{r_angle}{sup 1/2}. Global energy balance is discussed, and the coupled rate equations for {epsilon}(s) and R{sub b}(s) are examined for the class of axisymmetric beam distributions F{sub b}.
DEFF Research Database (Denmark)
Simonsen, Maria; Schiøler, Henrik; Leth, John-Josef
2014-01-01
The Euler-Maruyama method is applied to a simple stochastic differential equation (SDE) with discontinuous drift. Convergence aspects are investigated in the case, where the Euler-Maruyama method is simulated in dyadic points. A strong rate of convergence is presented for the numerical simulations...... and it is shown that the produced sequences converge almost surely. This is an improvement of the general result for SDEs with discontinuous drift, i.e. that the Euler-Maruyama approximations converge in probability to a strong solution of the SDE. A numerical example is presented together with a confidence...
Brzeźniak, Zdzisław; Motyl, Elżbieta
Stochastic Navier-Stokes equations in 2D and 3D possibly unbounded domains driven by a multiplicative Gaussian noise are considered. The noise term depends on the unknown velocity and its spatial derivatives. The existence of a martingale solution is proved. The construction of the solution is based on the classical Faedo-Galerkin approximation, the compactness method and the Jakubowski version of the Skorokhod Theorem for nonmetric spaces. Moreover, some compactness and tightness criteria in nonmetric spaces are proved. Compactness results are based on a certain generalization of the classical Dubinsky Theorem.
Numerical solutions of stochastic Lotka-Volterra equations via operational matrices
Directory of Open Access Journals (Sweden)
F. Hosseini Shekarabi
2016-03-01
Full Text Available In this paper, an efficient and convenient method for numerical solutions of stochastic Lotka-Volterra dynamical system is proposed. Here, we consider block pulse functions and their operational matrices of integration. Illustrative example is included to demonstrate the procedure and accuracy of the operational matrices based on block pulse functions.
International Nuclear Information System (INIS)
El-Tawil, M A; Al-Jihany, A S
2008-01-01
In this paper, nonlinear oscillators under quadratic nonlinearity with stochastic inputs are considered. Different methods are used to obtain first order approximations, namely, the WHEP technique, the perturbation method, the Pickard approximations, the Adomian decompositions and the homotopy perturbation method (HPM). Some statistical moments are computed for the different methods using mathematica 5. Comparisons are illustrated through figures for different case-studies
Avanzini, Francesco; Moro, Giorgio J
2018-03-15
The quantum molecular trajectory is the deterministic trajectory, arising from the Bohm theory, that describes the instantaneous positions of the nuclei of molecules by assuring the agreement with the predictions of quantum mechanics. Therefore, it provides the suitable framework for representing the geometry and the motions of molecules without neglecting their quantum nature. However, the quantum molecular trajectory is extremely demanding from the computational point of view, and this strongly limits its applications. To overcome such a drawback, we derive a stochastic representation of the quantum molecular trajectory, through projection operator techniques, for the degrees of freedom of an open quantum system. The resulting Fokker-Planck operator is parametrically dependent upon the reduced density matrix of the open system. Because of the pilot role played by the reduced density matrix, this stochastic approach is able to represent accurately the main features of the open system motions both at equilibrium and out of equilibrium with the environment. To verify this procedure, the predictions of the stochastic and deterministic representation are compared for a model system of six interacting harmonic oscillators, where one oscillator is taken as the open quantum system of interest. The undeniable advantage of the stochastic approach is that of providing a simplified and self-contained representation of the dynamics of the open system coordinates. Furthermore, it can be employed to study the out of equilibrium dynamics and the relaxation of quantum molecular motions during photoinduced processes, like photoinduced conformational changes and proton transfers.
Directory of Open Access Journals (Sweden)
2006-01-01
Full Text Available We show the existence of a solution for the double-barrier reflected BSDE when the barriers are completely separate and the generator is continuous with quadratic growth. As an application, we solve the risk-sensitive mixed zero-sum stochastic differential game. In addition we deal with recallable options under Knightian uncertainty.
S. Hamadène; I. Hdhiri
2006-01-01
We show the existence of a solution for the double-barrier reflected BSDE when the barriers are completely separate and the generator is continuous with quadratic growth. As an application, we solve the risk-sensitive mixed zero-sum stochastic differential game. In addition we deal with recallable options under Knightian uncertainty.
Directory of Open Access Journals (Sweden)
I. Hdhiri
2006-02-01
Full Text Available We show the existence of a solution for the double-barrier reflected BSDE when the barriers are completely separate and the generator is continuous with quadratic growth. As an application, we solve the risk-sensitive mixed zero-sum stochastic differential game. In addition we deal with recallable options under Knightian uncertainty.
Parlar, Mahmut
2004-01-01
Brownian motion is an important stochastic process used in modelling the random evolution of stock prices. In their 1973 seminal paper--which led to the awarding of the 1997 Nobel prize in Economic Sciences--Fischer Black and Myron Scholes assumed that the random stock price process is described (i.e., generated) by Brownian motion. Despite its…
Mulac, Richard A.; Celestina, Mark L.; Adamczyk, John J.; Misegades, Kent P.; Dawson, Jef M.
1987-01-01
A procedure is outlined which utilizes parallel processing to solve the inviscid form of the average-passage equation system for multistage turbomachinery along with a description of its implementation in a FORTRAN computer code, MSTAGE. A scheme to reduce the central memory requirements of the program is also detailed. Both the multitasking and I/O routines referred to in this paper are specific to the Cray X-MP line of computers and its associated SSD (Solid-state Storage Device). Results are presented for a simulation of a two-stage rocket engine fuel pump turbine.
Mulac, Richard A.; Celestina, Mark L.; Adamczyk, John J.; Misegades, Kent P.; Dawson, Jef M.
1987-01-01
A procedure is outlined which utilizes parallel processing to solve the inviscid form of the average-passage equation system for multistage turbomachinery along with a description of its implementation in a FORTRAN computer code, MSTAGE. A scheme to reduce the central memory requirements of the program is also detailed. Both the multitasking and I/O routines referred to are specific to the Cray X-MP line of computers and its associated SSD (Solid-State Disk). Results are presented for a simulation of a two-stage rocket engine fuel pump turbine.
DEFF Research Database (Denmark)
Lodi, C.; Bacher, Peder; Cipriano, J.
2012-01-01
and heat transfer coefficients is fundamental in order to improve the thermo-electrical production.The considered grey-box models are composed of a set of continuous time stochastic differential equations, holding the physical description of the system, combined with a set of discrete time measurement...... reduce the ventilation thermal losses of the building by pre-heating the fresh air. Furthermore, by decreasing PV module temperature, the ventilation air heat extraction can simultaneously increase electrical and thermal energy production of the building. A correct prediction of the PV module temperature...... equations, which represent the data driven part.In the present work, both one-state and two-state non-linear grey-box models are considered. In order to validate the results, the residuals are analysed for white-noise properties....
Fractional Stochastic Field Theory
Honkonen, Juha
2018-02-01
Models describing evolution of physical, chemical, biological, social and financial processes are often formulated as differential equations with the understanding that they are large-scale equations for averages of quantities describing intrinsically random processes. Explicit account of randomness may lead to significant changes in the asymptotic behaviour (anomalous scaling) in such models especially in low spatial dimensions, which in many cases may be captured with the use of the renormalization group. Anomalous scaling and memory effects may also be introduced with the use of fractional derivatives and fractional noise. Construction of renormalized stochastic field theory with fractional derivatives and fractional noise in the underlying stochastic differential equations and master equations and the interplay between fluctuation-induced and built-in anomalous scaling behaviour is reviewed and discussed.
Workshop on quantum stochastic differential equations for the quantum simulation of physical systems
2016-09-22
and a milestone as ARL becomes a significant player in the development of mathematical tools for quantum systems of interest to the Army...Commun. Math . Phys. 104, 457 (1986). [5] M.-H. Chang, Quantum stochastics, Cambridge Series in Statistical and Probabilistic Mathematics (2014...Phys. A 28, 5401 (1995). [9] M. J. Kastoryano and K. Temme, Quantum logarithmic Sobolev inequalities and rapid mixing, J. Math . Phys. 54, 052202
Ergodicity of stochastic 2D Navier-Stokes equation with Lévy noise
Dong, Zhao; Xie, Yingchao
In this paper we deal with the 2D Navier-Stokes equation perturbed by a Lévy noise force whose white noise part is non-degenerate and that the intensity measure of Poisson measure is σ-finite. Existence and uniqueness of invariant measure for this equation is obtained, two main properties of the Markov semigroup associated with this equation are proved. In other words, strong Feller property and irreducibility hold in the same space.
Scott, JoAnna M; deCamp, Allan; Juraska, Michal; Fay, Michael P; Gilbert, Peter B
2017-04-01
Stepped wedge designs are increasingly commonplace and advantageous for cluster randomized trials when it is both unethical to assign placebo, and it is logistically difficult to allocate an intervention simultaneously to many clusters. We study marginal mean models fit with generalized estimating equations for assessing treatment effectiveness in stepped wedge cluster randomized trials. This approach has advantages over the more commonly used mixed models that (1) the population-average parameters have an important interpretation for public health applications and (2) they avoid untestable assumptions on latent variable distributions and avoid parametric assumptions about error distributions, therefore, providing more robust evidence on treatment effects. However, cluster randomized trials typically have a small number of clusters, rendering the standard generalized estimating equation sandwich variance estimator biased and highly variable and hence yielding incorrect inferences. We study the usual asymptotic generalized estimating equation inferences (i.e., using sandwich variance estimators and asymptotic normality) and four small-sample corrections to generalized estimating equation for stepped wedge cluster randomized trials and for parallel cluster randomized trials as a comparison. We show by simulation that the small-sample corrections provide improvement, with one correction appearing to provide at least nominal coverage even with only 10 clusters per group. These results demonstrate the viability of the marginal mean approach for both stepped wedge and parallel cluster randomized trials. We also study the comparative performance of the corrected methods for stepped wedge and parallel designs, and describe how the methods can accommodate interval censoring of individual failure times and incorporate semiparametric efficient estimators.
International Nuclear Information System (INIS)
Keanini, R.G.
2011-01-01
Research highlights: → Systematic approach for physically probing nonlinear and random evolution problems. → Evolution of vortex sheets corresponds to evolution of an Ornstein-Uhlenbeck process. → Organization of near-molecular scale vorticity mediated by hydrodynamic modes. → Framework allows calculation of vorticity evolution within random strain fields. - Abstract: A framework which combines Green's function (GF) methods and techniques from the theory of stochastic processes is proposed for tackling nonlinear evolution problems. The framework, established by a series of easy-to-derive equivalences between Green's function and stochastic representative solutions of linear drift-diffusion problems, provides a flexible structure within which nonlinear evolution problems can be analyzed and physically probed. As a preliminary test bed, two canonical, nonlinear evolution problems - Burgers' equation and the nonlinear Schroedinger's equation - are first treated. In the first case, the framework provides a rigorous, probabilistic derivation of the well known Cole-Hopf ansatz. Likewise, in the second, the machinery allows systematic recovery of a known soliton solution. The framework is then applied to a fairly extensive exploration of physical features underlying evolution of randomly stretched and advected Burger's vortex sheets. Here, the governing vorticity equation corresponds to the Fokker-Planck equation of an Ornstein-Uhlenbeck process, a correspondence that motivates an investigation of sub-sheet vorticity evolution and organization. Under the assumption that weak hydrodynamic fluctuations organize disordered, near-molecular-scale, sub-sheet vorticity, it is shown that these modes consist of two weakly damped counter-propagating cross-sheet acoustic modes, a diffusive cross-sheet shear mode, and a diffusive cross-sheet entropy mode. Once a consistent picture of in-sheet vorticity evolution is established, a number of analytical results, describing the
Dolgov, Sergey
2015-11-03
We apply the tensor train (TT) decomposition to construct the tensor product polynomial chaos expansion (PCE) of a random field, to solve the stochastic elliptic diffusion PDE with the stochastic Galerkin discretization, and to compute some quantities of interest (mean, variance, and exceedance probabilities). We assume that the random diffusion coefficient is given as a smooth transformation of a Gaussian random field. In this case, the PCE is delivered by a complicated formula, which lacks an analytic TT representation. To construct its TT approximation numerically, we develop the new block TT cross algorithm, a method that computes the whole TT decomposition from a few evaluations of the PCE formula. The new method is conceptually similar to the adaptive cross approximation in the TT format but is more efficient when several tensors must be stored in the same TT representation, which is the case for the PCE. In addition, we demonstrate how to assemble the stochastic Galerkin matrix and to compute the solution of the elliptic equation and its postprocessing, staying in the TT format. We compare our technique with the traditional sparse polynomial chaos and the Monte Carlo approaches. In the tensor product polynomial chaos, the polynomial degree is bounded for each random variable independently. This provides higher accuracy than the sparse polynomial set or the Monte Carlo method, but the cardinality of the tensor product set grows exponentially with the number of random variables. However, when the PCE coefficients are implicitly approximated in the TT format, the computations with the full tensor product polynomial set become possible. In the numerical experiments, we confirm that the new methodology is competitive in a wide range of parameters, especially where high accuracy and high polynomial degrees are required.
Directory of Open Access Journals (Sweden)
V. Lucarini
2012-01-01
Full Text Available The understanding of the statistical properties and of the dynamics of multistable systems is gaining more and more importance in a vast variety of scientific fields. This is especially relevant for the investigation of the tipping points of complex systems. Sometimes, in order to understand the time series of given observables exhibiting bimodal distributions, simple one-dimensional Langevin models are fitted to reproduce the observed statistical properties, and used to investing-ate the projected dynamics of the observable. This is of great relevance for studying potential catastrophic changes in the properties of the underlying system or resonant behaviours like those related to stochastic resonance-like mechanisms. In this paper, we propose a framework for encasing this kind of studies, using simple box models of the oceanic circulation and choosing as observable the strength of the thermohaline circulation. We study the statistical properties of the transitions between the two modes of operation of the thermohaline circulation under symmetric boundary forcings and test their agreement with simplified one-dimensional phenomenological theories. We extend our analysis to include stochastic resonance-like amplification processes. We conclude that fitted one-dimensional Langevin models, when closely scrutinised, may result to be more ad-hoc than they seem, lacking robustness and/or well-posedness. They should be treated with care, more as an empiric descriptive tool than as methodology with predictive power.
Random Attractors for the Stochastic Navier-Stokes Equations on the 2D Unit Sphere
Brzeźniak, Z.; Goldys, B.; Le Gia, Q. T.
2018-03-01
In this paper we prove the existence of random attractors for the Navier-Stokes equations on 2 dimensional sphere under random forcing irregular in space and time. We also deduce the existence of an invariant measure.
Johnson, Paul; Howell, Sydney; Duck, Peter
2017-08-13
A mixed financial/physical partial differential equation (PDE) can optimize the joint earnings of a single wind power generator (WPG) and a generic energy storage device (ESD). Physically, the PDE includes constraints on the ESD's capacity, efficiency and maximum speeds of charge and discharge. There is a mean-reverting daily stochastic cycle for WPG power output. Physically, energy can only be produced or delivered at finite rates. All suppliers must commit hourly to a finite rate of delivery C , which is a continuous control variable that is changed hourly. Financially, we assume heavy 'system balancing' penalties in continuous time, for deviations of output rate from the commitment C Also, the electricity spot price follows a mean-reverting stochastic cycle with a strong evening peak, when system balancing penalties also peak. Hence the economic goal of the WPG plus ESD, at each decision point, is to maximize expected net present value (NPV) of all earnings (arbitrage) minus the NPV of all expected system balancing penalties, along all financially/physically feasible future paths through state space. Given the capital costs for the various combinations of the physical parameters, the design and operating rules for a WPG plus ESD in a finite market may be jointly optimizable.This article is part of the themed issue 'Energy management: flexibility, risk and optimization'. © 2017 The Author(s).
International Nuclear Information System (INIS)
Basharov, A. M.
2012-01-01
It is shown that the effective Hamiltonian representation, as it is formulated in author’s papers, serves as a basis for distinguishing, in a broadband environment of an open quantum system, independent noise sources that determine, in terms of the stationary quantum Wiener and Poisson processes in the Markov approximation, the effective Hamiltonian and the equation for the evolution operator of the open system and its environment. General stochastic differential equations of generalized Langevin (non-Wiener) type for the evolution operator and the kinetic equation for the density matrix of an open system are obtained, which allow one to analyze the dynamics of a wide class of localized open systems in the Markov approximation. The main distinctive features of the dynamics of open quantum systems described in this way are the stabilization of excited states with respect to collective processes and an additional frequency shift of the spectrum of the open system. As an illustration of the general approach developed, the photon dynamics in a single-mode cavity without losses on the mirrors is considered, which contains identical intracavity atoms coupled to the external vacuum electromagnetic field. For some atomic densities, the photons of the cavity mode are “locked” inside the cavity, thus exhibiting a new phenomenon of radiation trapping and non-Wiener dynamics.
Belkina, T. A.; Konyukhova, N. B.; Kurochkin, S. V.
2012-10-01
A singular boundary value problem for a second-order linear integrodifferential equation with Volterra and non-Volterra integral operators is formulated and analyzed. The equation is defined on ℝ+, has a weak singularity at zero and a strong singularity at infinity, and depends on several positive parameters. Under natural constraints on the coefficients of the equation, existence and uniqueness theorems for this problem with given limit boundary conditions at singular points are proved, asymptotic representations of the solution are given, and an algorithm for its numerical determination is described. Numerical computations are performed and their interpretation is given. The problem arises in the study of the survival probability of an insurance company over infinite time (as a function of its initial surplus) in a dynamic insurance model that is a modification of the classical Cramer-Lundberg model with a stochastic process rate of premium under a certain investment strategy in the financial market. A comparative analysis of the results with those produced by the model with deterministic premiums is given.
Strauss, R. Du Toit; Effenberger, Frederic
2017-10-01
In this review, an overview of the recent history of stochastic differential equations (SDEs) in application to particle transport problems in space physics and astrophysics is given. The aim is to present a helpful working guide to the literature and at the same time introduce key principles of the SDE approach via "toy models". Using these examples, we hope to provide an easy way for newcomers to the field to use such methods in their own research. Aspects covered are the solar modulation of cosmic rays, diffusive shock acceleration, galactic cosmic ray propagation and solar energetic particle transport. We believe that the SDE method, due to its simplicity and computational efficiency on modern computer architectures, will be of significant relevance in energetic particle studies in the years to come.
Faizullah, Faiz
2016-01-01
The aim of the current paper is to present the path-wise and moment estimates for solutions to stochastic functional differential equations with non-linear growth condition in the framework of G-expectation and G-Brownian motion. Under the nonlinear growth condition, the pth moment estimates for solutions to SFDEs driven by G-Brownian motion are proved. The properties of G-expectations, Hölder's inequality, Bihari's inequality, Gronwall's inequality and Burkholder-Davis-Gundy inequalities are used to develop the above mentioned theory. In addition, the path-wise asymptotic estimates and continuity of pth moment for the solutions to SFDEs in the G-framework, with non-linear growth condition are shown.
Passive tracer in a flow corresponding to two-dimensional stochastic Navier–Stokes equations
International Nuclear Information System (INIS)
Komorowski, Tomasz; Peszat, Szymon; Szarek, Tomasz
2013-01-01
In this paper we prove the law of large numbers and central limit theorem for trajectories of a particle carried by a two-dimensional Eulerian velocity field. The field is given by a solution of a stochastic Navier–Stokes system with non-degenerate noise. The spectral gap property, with respect to the Wasserstein metric, for such a system was shown in Hairer and Mattingly (2008 Ann. Probab. 36 2050–91). In this paper we show that a similar property holds for the environment process corresponding to the Lagrangian observations of the velocity. Consequently we conclude the law of large numbers and the central limit theorem for the tracer. The proof of the central limit theorem relies on the martingale approximation of the trajectory process. (paper)
A stochastic multiscale method for the elastodynamic wave equation arising from fiber composites
Babuška, Ivo
2014-03-21
We present a stochastic multilevel global–local algorithm for computing elastic waves propagating in fiber-reinforced composite materials. Here, the materials properties and the size and location of fibers may be random. The method aims at approximating statistical moments of some given quantities of interest, such as stresses, in regions of relatively small size, e.g. hot spots or zones that are deemed vulnerable to failure. For a fiber-reinforced cross-plied laminate, we introduce three problems (macro, meso, micro) corresponding to the three natural scales, namely the sizes of laminate, ply, and fiber. The algorithm uses the homogenized global solution to construct a good local approximation that captures the microscale features of the real solution. We perform numerical experiments to show the applicability and efficiency of the method.
A Stochastic Multiscale Method for the Elastic Wave Equations Arising from Fiber Composites
Babuska, Ivo
2016-01-06
We present a stochastic multilevel global-local algorithm [1] for computing elastic waves propagating in fiber-reinforced polymer composites, where the material properties and the size and distribution of fibers in the polymer matrix may be random. The method aims at approximating statistical moments of some given quantities of interest, such as stresses, in regions of relatively small size, e.g. hot spots or zones that are deemed vulnerable to failure. For a fiber-reinforced cross-plied laminate, we introduce three problems: 1) macro; 2) meso; and 3) micro problems, corresponding to the three natural length scales: 1) the sizes of plate; 2) the tickles of plies; and 3) and the diameter of fibers. The algorithm uses a homogenized global solution to construct a local approximation that captures the microscale features of the problem. We perform numerical experiments to show the applicability and efficiency of the method.
Shiino, Masatoshi; Yamana, Michiko
2004-01-01
We study the statistical mechanical aspects of stochastic analog neural network models for associative memory with correlation type learning. We take three approaches to derive the set of the order parameter equations for investigating statistical properties of retrieval states: the self-consistent signal-to-noise analysis (SCSNA), the Thouless-Anderson-Palmer (TAP) equation, and the replica symmetric calculation. On the basis of the cavity method the SCSNA can be generalized to deal with stochastic networks. We establish the close connection between the TAP equation and the SCSNA to elucidate the relationship between the Onsager reaction term of the TAP equation and the output proportional term of the SCSNA that appear in the expressions for the local fields.
McKean, Henry P
2005-01-01
This little book is a brilliant introduction to an important boundary field between the theory of probability and differential equations. -E. B. Dynkin, Mathematical Reviews This well-written book has been used for many years to learn about stochastic integrals. The book starts with the presentation of Brownian motion, then deals with stochastic integrals and differentials, including the famous Itô lemma. The rest of the book is devoted to various topics of stochastic integral equations, including those on smooth manifolds. Originally published in 1969, this classic book is ideal for supplemen
Directory of Open Access Journals (Sweden)
Maxim Ioan
2009-05-01
Full Text Available In our paper we build a reccurence from generalized Garman equation and discretization of 3-dimensional domain. From reccurence we build an algorithm for computing values of an option based on time, momentan volatility of support and value of support on a
Cuenod, Charles-André; Favetto, Benjamin; Genon-Catalot, Valentine; Rozenholc, Yves; Samson, Adeline
2011-09-01
Dynamic Contrast Enhanced imaging (DCE-imaging) following a contrast agent bolus allows the extraction of information on tissue micro-vascularization. The dynamic signals obtained from DCE-imaging are modeled by pharmacokinetic compartmental models which integrate the Arterial Input Function. These models use ordinary differential equations (ODEs) to describe the exchanges between the arterial and capillary plasma and the extravascular-extracellular space. Their least squares fitting takes into account measurement noises but fails to deal with unpredictable fluctuations due to external/internal sources of variations (patients' anxiety, time-varying parameters, measurement errors in the input function, etc.). Adding Brownian components to the ODEs leads to stochastic differential equations (SDEs). In DCE-imaging, SDEs are discretely observed with an additional measurement noise. We propose to estimate the parameters of these noisy SDEs by maximum likelihood, using the Kalman filter. In DCE-imaging, the contrast agent injected in vein arrives in plasma with an unknown time delay. The delay parameter induces a change-point in the drift of the SDE and ODE models, which is estimated also. Estimations based on the SDE and ODE pharmacokinetic models are compared to real DCE-MRI data. They show that the use of SDE provides robustness in the estimation results. A simulation study confirms these results. Copyright © 2011 Elsevier Inc. All rights reserved.
Overgaard, Rune Viig; Holford, Nick; Rytved, Klaus A; Madsen, Henrik
2007-02-01
To describe the pharmacodynamic effects of recombinant human interleukin-21 (IL-21) on core body temperature in cynomolgus monkeys using basic mechanisms of heat regulation. A major effort was devoted to compare the use of ordinary differential equations (ODEs) with stochastic differential equations (SDEs) in pharmacokinetic pharmacodynamic (PKPD) modelling. A temperature model was formulated including circadian rhythm, metabolism, heat loss, and a thermoregulatory set-point. This model was formulated as a mixed-effects model based on SDEs using NONMEM. The effects of IL-21 were on the set-point and the circadian rhythm of metabolism. The model was able to describe a complex set of IL-21 induced phenomena, including 1) disappearance of the circadian rhythm, 2) no effect after first dose, and 3) high variability after second dose. SDEs provided a more realistic description with improved simulation properties, and further changed the model into one that could not be falsified by the autocorrelation function. The IL-21 induced effects on thermoregulation in cynomolgus monkeys are explained by a biologically plausible model. The quality of the model was improved by the use of SDEs.
Silveri, M.; Zalys-Geller, E.; Hatridge, M.; Leghtas, Z.; Devoret, M. H.; Girvin, S. M.
2015-03-01
In the remote entanglement process, two distant stationary qubits are entangled with separate flying qubits and the which-path information is erased from the flying qubits by interference effects. As a result, an observer cannot tell from which of the two sources a signal came and the probabilistic measurement process generates perfect heralded entanglement between the two signal sources. Notably, the two stationary qubits are spatially separated and there is no direct interaction between them. We study two transmon qubits in superconducting cavities connected to a Josephson Parametric Converter (JPC). The qubit information is encoded in the traveling wave leaking out from each cavity. Remarkably, the quantum-limited phase-preserving amplification of two traveling waves provided by the JPC can work as a which-path information eraser. By using a stochastic master approach we demonstrate the probabilistic production of heralded entangled states and that unequal qubit-cavity pairs can be made indistinguishable by simple engineering of driving fields. Additionally, we will derive measurement rates, measurement optimization strategies and discuss the effects of finite amplification gain, cavity losses, and qubit relaxations and dephasing. Work supported by IARPA, ARO and NSF.
Energy Technology Data Exchange (ETDEWEB)
Decker, J.; Peysson, Y
2004-12-01
A new original code for solving the 3-D relativistic and bounce-averaged electron drift kinetic equation is presented. It designed for the current drive problem in tokamak with an arbitrary magnetic equilibrium. This tool allows self-consistent calculations of the bootstrap current in presence of other external current sources. RF current drive for arbitrary type of waves may be used. Several moments of the electron distribution function are determined, like the exact and effective fractions of trapped electrons, the plasma current, absorbed RF power, runaway and magnetic ripple loss rates and non-thermal Bremsstrahlung. Advanced numerical techniques have been used to make it the first fully implicit (reverse time) 3-D solver, particularly well designed for implementation in a chain of code for realistic current drive calculations in high {beta}{sub p} plasmas. All the details of the physics background and the numerical scheme are presented, as well a some examples to illustrate main code capabilities. Several important numerical points are addressed concerning code stability and potential numerical and physical limitations. (authors)
SITU, Rong
2005-01-01
Derivation of Ito's formulas, Girsanov's theorems and martingale representation theorem for stochastic DEs with jumpsApplications to population controlReflecting stochastic DE techniqueApplications to the stock market. (Backward stochastic DE approach)Derivation of Black-Scholes formula for market with and without jumpsNon-linear filtering problems with jumps.
Shushin, A I
2008-03-01
Some specific features and extensions of the continuous-time random-walk (CTRW) approach are analyzed in detail within the Markovian representation (MR) and CTRW-based non-Markovian stochastic Liouville equation (SLE). In the MR, CTRW processes are represented by multidimensional Markovian ones. In this representation the probability density function (PDF) W(t) of fluctuation renewals is associated with that of reoccurrences in a certain jump state of some Markovian controlling process. Within the MR the non-Markovian SLE, which describes the effect of CTRW-like noise on the relaxation of dynamic and stochastic systems, is generalized to take into account the influence of relaxing systems on the statistical properties of noise. Some applications of the generalized non-Markovian SLE are discussed. In particular, it is applied to study two modifications of the CTRW approach. One of them considers cascaded CTRWs in which the controlling process is actually a CTRW-like one controlled by another CTRW process, controlled in turn by a third one, etc. Within the MR a simple expression for the PDF W(t) of the total controlling process is obtained in terms of Markovian variants of controlling PDFs in the cascade. The expression is shown to be especially simple and instructive in the case of anomalous processes determined by the long-time tailed W(t) . The cascaded CTRWs can model the effect of the complexity of a system on the relaxation kinetics (in glasses, fractals, branching media, ultrametric structures, etc.). Another CTRW modification describes the kinetics of processes governed by fluctuating W(t) . Within the MR the problem is analyzed in a general form without restrictive assumptions on the correlations of PDFs of consecutive renewals. The analysis shows that fluctuations of W(t) can strongly affect the kinetics of the process. Possible manifestations of this effect are discussed.
Wijsen, N.; Poedts, S.; Pomoell, J.
2017-12-01
Solar energetic particles (SEPs) are high energy particles originating from solar eruptive events. These particles can be energised at solar flare sites during magnetic reconnection events, or in shock waves propagating in front of coronal mass ejections (CMEs). These CME-driven shocks are in particular believed to act as powerful accelerators of charged particles throughout their propagation in the solar corona. After escaping from their acceleration site, SEPs propagate through the heliosphere and may eventually reach our planet where they can disrupt the microelectronics on satellites in orbit and endanger astronauts among other effects. Therefore it is of vital importance to understand and thereby build models capable of predicting the characteristics of SEP events. The propagation of SEPs in the heliosphere can be described by the time-dependent focused transport equation. This five-dimensional parabolic partial differential equation can be solved using e.g., a finite difference method or by integrating a set of corresponding first order stochastic differential equations. In this work we take the latter approach to model SEP events under different solar wind and scattering conditions. The background solar wind in which the energetic particles propagate is computed using a magnetohydrodynamic model. This allows us to study the influence of different realistic heliospheric configurations on SEP transport. In particular, in this study we focus on exploring the influence of high speed solar wind streams originating from coronal holes that are located close to the eruption source region on the resulting particle characteristics at Earth. Finally, we discuss our upcoming efforts towards integrating our particle propagation model with time-dependent heliospheric MHD space weather modelling.
Rare event simulation for processes generated via stochastic fixed point equations
DEFF Research Database (Denmark)
Collamore, Jeffrey F.; Diao, Guoqing; Vidyashankar, Anand N.
2014-01-01
In a number of applications, particularly in financial and actuarial mathematics, it is of interest to characterize the tail distribution of a random variable V satisfying the distributional equation V=_D f(V), for some random function f. This paper is concerned with computational methods...... for evaluating these tail probabilities. We introduce a novel importance sampling algorithm, involving an exponential shift over a random time interval, for estimating these rare event probabilities. We prove that the proposed estimator is: (i) consistent, (ii) strongly efficient and (iii) optimal within a wide...
International Nuclear Information System (INIS)
Appleby, J A D; Wu, H
2008-01-01
In this paper we consider functional differential equations subjected to either instantaneous state-dependent noise, or to a white noise perturbation. The drift of the equations depend linearly on the current value and on the maximum of the solution. The functional term always provides positive feedback, while the instantaneous term can be mean-reverting or can exhibit positive feedback. We show in the white noise case that if the instantaneous term is mean reverting and dominates the history term, then solutions are recurrent, and upper bounds on the a.s. growth rate of the partial maxima of the solution can be found. When the instantaneous term is weaker, or is of positive feedback type, we determine necessary and sufficient conditions on the diffusion coefficient which ensure the exact exponential growth of solutions. An application of these results to an inefficient financial market populated by reference traders and speculators is given, in which the difference between the current instantaneous returns and maximum of the returns over the last few time units is used to determine trading strategies.
Probability and Cumulative Density Function Methods for the Stochastic Advection-Reaction Equation
Energy Technology Data Exchange (ETDEWEB)
Barajas-Solano, David A.; Tartakovsky, Alexandre M.
2018-01-01
We present a cumulative density function (CDF) method for the probabilistic analysis of $d$-dimensional advection-dominated reactive transport in heterogeneous media. We employ a probabilistic approach in which epistemic uncertainty on the spatial heterogeneity of Darcy-scale transport coefficients is modeled in terms of random fields with given correlation structures. Our proposed CDF method employs a modified Large-Eddy-Diffusivity (LED) approach to close and localize the nonlocal equations governing the one-point PDF and CDF of the concentration field, resulting in a $(d + 1)$ dimensional PDE. Compared to the classsical LED localization, the proposed modified LED localization explicitly accounts for the mean-field advective dynamics over the phase space of the PDF and CDF. To illustrate the accuracy of the proposed closure, we apply our CDF method to one-dimensional single-species reactive transport with uncertain, heterogeneous advection velocities and reaction rates modeled as random fields.
Cu(II)-porphyrin molecular dynamics as seen in a novel EPR/Stochastic Liouville equation study.
Håkansson, Pär; Nguyen, ThaoNguyen; Nair, Prasanth B; Edge, Ruth; Stulz, Eugen
2013-07-14
Copper porphyrin dissolved in CH2Cl2:toluene as fluid and frozen solution was studied as a function of temperature using X-band electron paramagnetic resonance (EPR). Quantitative interpretation was obtained using a recently developed Stochastic Liouville simulation method. For the first time we address the large spin system that translates into a 400,000 dimensional Liouville equation solved under slow-motion conditions. Using a simple three parameter microscopic model, the temperature dependence of porphyrin rotational correlation time is determined to be in the range 1-10 ns and a fast local motion is in the subpico-second regime with an amplitude increasing with temperature. The methodology provides an important tool for arriving at an accurate set of spin Hamiltonian parameters since determining a unique set of parameters from a frozen solution EPR experiment is often difficult. Thus, the proposed method discriminates between parameters proposed from frozen solution EPR experiments or quantum chemistry calculations. The methodology presented is expected to be valuable in obtaining a molecular dynamics picture of metal proteins using EPR as well as in the study of artificial photosynthetic systems.
International Nuclear Information System (INIS)
Neate, A D; Truman, A
2005-01-01
The inviscid limit of the Burgers equation, with body forces white noise in time, is discussed in terms of the level surfaces of the minimizing Hamilton-Jacobi function and the classical mechanical caustic and their algebraic pre-images under the classical mechanical flow map. The problem is analysed in terms of a reduced (one-dimensional) action function using a circle of ideas due to Arnol'd, Cayley and Klein. We characterize those parts of the caustic which are singular, and give an explicit expression for the cusp density on caustics and level surfaces. By considering the double points of level surfaces we find an explicit formula for the Maxwell set in the two-dimensional polynomial case, and we extend this to higher dimensions using a double discriminant of the reduced action, solving a long-standing problem for Hamiltonian dynamical systems. When the pre-level surface touches the pre-caustic, the geometry (number of cusps) on the level surface changes infinitely rapidly causing 'real turbulence'. Using an idea of Klein, it is shown that the geometry (number of swallowtails) on the caustic also changes infinitely rapidly when the real part of the pre-caustic touches its complex counterpart, causing 'complex turbulence'. These are both inherently stochastic in nature, and we determine their intermittence in terms of the recurrent behaviour of two processes
Mortensen, Stig B; Klim, Søren; Dammann, Bernd; Kristensen, Niels R; Madsen, Henrik; Overgaard, Rune V
2007-10-01
The non-linear mixed-effects model based on stochastic differential equations (SDEs) provides an attractive residual error model, that is able to handle serially correlated residuals typically arising from structural mis-specification of the true underlying model. The use of SDEs also opens up for new tools for model development and easily allows for tracking of unknown inputs and parameters over time. An algorithm for maximum likelihood estimation of the model has earlier been proposed, and the present paper presents the first general implementation of this algorithm. The implementation is done in Matlab and also demonstrates the use of parallel computing for improved estimation times. The use of the implementation is illustrated by two examples of application which focus on the ability of the model to estimate unknown inputs facilitated by the extension to SDEs. The first application is a deconvolution-type estimation of the insulin secretion rate based on a linear two-compartment model for C-peptide measurements. In the second application the model is extended to also give an estimate of the time varying liver extraction based on both C-peptide and insulin measurements.
Ullah, Rahman; Faizullah, Faiz
2017-10-01
This investigation aims at studying a Euler-Maruyama (EM) approximate solutions scheme for stochastic differential equations (SDEs) in the framework of G-Brownian motion. Subject to the growth condition, it is shown that the EM solutions Z^q(t) are bounded, in particular, Z^q(t)\\in M_G^2([t_0,T];R^n) . Letting Z( t) as a unique solution to SDEs in the G-framework and utilizing the growth and Lipschitz conditions, the convergence of Z^q(t) to Z( t) is revealed. The Burkholder-Davis-Gundy (BDG) inequalities, Hölder's inequality, Gronwall's inequality and Doobs martingale's inequality are used to derive the results. In addition, without assuming a solution of the stated SDE, we have shown that the Euler-Maruyama approximation sequence {Z^q(t)} is Cauchy in M_G^2([t_0,T];R^n) thus converges to a limit which is a unique solution to SDE in the G-framework.
Stochastic stability of four-wheel-steering system
International Nuclear Information System (INIS)
Huang Dongwei; Wang Hongli; Zhu Zhiwen; Feng Zhang
2007-01-01
A four-wheel-steering system subjected to white noise excitations was reduced to a two-degree-of-freedom quasi-non-integrable-Hamiltonian system. Subsequently we obtained an one-dimensional Ito stochastic differential equation for the averaged Hamiltonian of the system by using the stochastic averaging method for quasi-non-integrable-Hamiltonian systems. Thus, the stochastic stability of four-wheel-steering system was analyzed by analyzing the sample behaviors of the averaged Hamiltonian at the boundary H = 0 and calculating its Lyapunov exponent. An example given at the end demonstrated that the conclusion obtained is of considerable significance
Stochastic stability of mechanical systems under renewal jump process parametric excitation
DEFF Research Database (Denmark)
Iwankiewicz, R.; Nielsen, Søren R.K.; Larsen, Jesper Winther
2005-01-01
independent, negative exponential distributed variables; hence, the arrival process may be termed as a generalized Erlang renewal process. The excitation process is governed by the stochastic equation driven by two independent Poisson processes, with different parameters. If the response in a single mode...... is investigated, the problem is governed in the state space by two stochastic equations, because the stochastic equation for the excitation process is autonomic. However due to the parametric nature of the excitation, the nonlinear term appears at the right-hand sides of the equations. The equations become linear...... of the stochastic equation governing the natural logarithm of the hyperspherical amplitude process and using the modification of the method wherein the time averaging of the pertinent expressions is replaced by ensemble averaging. It is found that the direct simulation is more suitable and that the asymptotic mean...
Leander, Jacob; Lundh, Torbjörn; Jirstrand, Mats
2014-05-01
In this paper we consider the problem of estimating parameters in ordinary differential equations given discrete time experimental data. The impact of going from an ordinary to a stochastic differential equation setting is investigated as a tool to overcome the problem of local minima in the objective function. Using two different models, it is demonstrated that by allowing noise in the underlying model itself, the objective functions to be minimized in the parameter estimation procedures are regularized in the sense that the number of local minima is reduced and better convergence is achieved. The advantage of using stochastic differential equations is that the actual states in the model are predicted from data and this will allow the prediction to stay close to data even when the parameters in the model is incorrect. The extended Kalman filter is used as a state estimator and sensitivity equations are provided to give an accurate calculation of the gradient of the objective function. The method is illustrated using in silico data from the FitzHugh-Nagumo model for excitable media and the Lotka-Volterra predator-prey system. The proposed method performs well on the models considered, and is able to regularize the objective function in both models. This leads to parameter estimation problems with fewer local minima which can be solved by efficient gradient-based methods. Copyright © 2014 The Authors. Published by Elsevier Inc. All rights reserved.
Energy Technology Data Exchange (ETDEWEB)
Garcia-Olivares, R. A.; Munoz Roldan, A.
1992-07-01
This document describes the use of a library of programs able to solve stochastic Fokker-Planck equations in a N-dimensional space. The input data are essentially: (i) the initial distribution of the stochastic variable, (ii) the drift and fluctuation coefficients as a function of the state (which can be obtained from the transition probabilities between neighboring states) and (iii) some parameters controlling the run. A last version of the library accepts sources and sinks defined in the states space. The output is the temporal evolution of the probability distribution in the space defined by a N-dimensional grid. Some applications and readings in Synergetic, Self-Organization, transport phenomena, Ecology and other fields are suggested. If the probability distribution is interpreted as a distribution of particles then the codes can be used to solve the N-dimensional problem of advection-diffusion. (Author) 16 refs.
Møller, Jonas B; Overgaard, Rune V; Madsen, Henrik; Hansen, Torben; Pedersen, Oluf; Ingwersen, Steen H
2010-02-01
Several articles have investigated stochastic differential equations (SDEs) in PK/PD models, but few have quantitatively investigated the benefits to predictive performance of models based on real data. Estimation of first phase insulin secretion which reflects beta-cell function using models of the OGTT is a difficult problem in need of further investigation. The present work aimed at investigating the power of SDEs to predict the first phase insulin secretion (AIR (0-8)) in the IVGTT based on parameters obtained from the minimal model of the OGTT, published by Breda et al. (Diabetes 50(1):150-158, 2001). In total 174 subjects underwent both an OGTT and a tolbutamide modified IVGTT. Estimation of parameters in the oral minimal model (OMM) was performed using the FOCE-method in NONMEM VI on insulin and C-peptide measurements. The suggested SDE models were based on a continuous AR(1) process, i.e. the Ornstein-Uhlenbeck process, and the extended Kalman filter was implemented in order to estimate the parameters of the models. Inclusion of the Ornstein-Uhlenbeck (OU) process caused improved description of the variation in the data as measured by the autocorrelation function (ACF) of one-step prediction errors. A main result was that application of SDE models improved the correlation between the individual first phase indexes obtained from OGTT and AIR (0-8) (r = 0.36 to r = 0.49 and r = 0.32 to r = 0.47 with C-peptide and insulin measurements, respectively). In addition to the increased correlation also the properties of the indexes obtained using the SDE models more correctly assessed the properties of the first phase indexes obtained from the IVGTT. In general it is concluded that the presented SDE approach not only caused autocorrelation of errors to decrease but also improved estimation of clinical measures obtained from the glucose tolerance tests. Since, the estimation time of extended models was not heavily increased compared to basic models, the applied method
Stochastic quantization and supersymmetry
International Nuclear Information System (INIS)
Kirschner, R.
1984-04-01
In the last years interest in stochastic quantization has increased. The method of quantization by stochastic relaxation processes has been proposed by Parisi and Wu, inspired by the extensive application of Monte Carlo simulations to quantum systems. Starting with the classical equations of motion of the system (field theory) and adding random force terms - the random force obeys a Gaussian distribution (white noise) - stochastic differential equations are obtained, in this context called Langevin equations, which are a central object in the theory of stochastic processes. (author)
Stochastic quantization of instantons
International Nuclear Information System (INIS)
Grandati, Y.; Berard, A.; Grange, P.
1996-01-01
The method of Parisi and Wu to quantize classical fields is applied to instanton solutions var-phi I of euclidian non-linear theory in one dimension. The solution var-phi var-epsilon of the corresponding Langevin equation is built through a singular perturbative expansion in var-epsilon=h 1/2 in the frame of the center of the mass of the instanton, where the difference var-phi var-epsilon -var-phi I carries only fluctuations of the instanton form. The relevance of the method is shown for the stochastic K dV equation with uniform noise in space: the exact solution usually obtained by the inverse scattering method is retrieved easily by the singular expansion. A general diagrammatic representation of the solution is then established which makes a thorough use of regrouping properties of stochastic diagrams derived in scalar field theory. Averaging over the noise and in the limit of infinite stochastic time, the authors obtain explicit expressions for the first two orders in var-epsilon of the pertrubed instanton of its Green function. Specializing to the Sine-Gordon and var-phi 4 models, the first anaharmonic correction is obtained analytically. The calculation is carried to second order for the var-phi 4 model, showing good convergence. 21 refs., 5 fig
Venturi, D.; Karniadakis, G. E.
2012-08-01
By using functional integral methods we determine new evolution equations satisfied by the joint response-excitation probability density function (PDF) associated with the stochastic solution to first-order nonlinear partial differential equations (PDEs). The theory is presented for both fully nonlinear and for quasilinear scalar PDEs subject to random boundary conditions, random initial conditions or random forcing terms. Particular applications are discussed for the classical linear and nonlinear advection equations and for the advection-reaction equation. By using a Fourier-Galerkin spectral method we obtain numerical solutions of the proposed response-excitation PDF equations. These numerical solutions are compared against those obtained by using more conventional statistical approaches such as probabilistic collocation and multi-element probabilistic collocation methods. It is found that the response-excitation approach yields accurate predictions of the statistical properties of the system. In addition, it allows to directly ascertain the tails of probabilistic distributions, thus facilitating the assessment of rare events and associated risks. The computational cost of the response-excitation method is order magnitudes smaller than the one of more conventional statistical approaches if the PDE is subject to high-dimensional random boundary or initial conditions. The question of high-dimensionality for evolution equations involving multidimensional joint response-excitation PDFs is also addressed.
Stochastic multi-symplectic Runge-Kutta methods for stochastic Hamiltonian PDEs
Zhang, Liying; Ji, Lihai
2018-01-01
In this paper, we consider stochastic Runge-Kutta methods for stochastic Hamiltonian partial differential equations and present some sufficient conditions for multisymplecticity of stochastic Runge-Kutta methods of stochastic Hamiltonian partial differential equations. Particularly, we apply these ideas to stochastic Maxwell equations with multiplicative noise, possessing the stochastic multi-symplectic conservation law and energy conservation law. Theoretical analysis shows that the methods ...
Gromme, Sherman; Mankinen, Edward A.; Prévot, Michel
2010-01-01
We present here the complete paleomagnetic laboratory results from a collection of approximately 1500 oriented cores from all 16 of the Galapagos Islands, Ecuador, collected by Allan Cox in 1964–1965 but nearly all previously unpublished. The islands are located in the eastern Pacific Ocean within 1.4° of latitude from the equator and range in age from historically erupted to 3 Ma, mostly determined by published K-Ar and 3He isotopic dating. The number of sites collected on each island ranges from 1 to 28, for a total of 186. After combining duplicate site mean directions, 149 are used for an overall mean direction and 8 represent excursions and one reversal path. Divided by geomagnetic polarity chron, 110 site means are Brunhes or Jaramillo (normal polarity), 27 are Matuyama (reversed polarity), and 12 are Gauss (both polarities). We have completed the magnetic cleaning that was commenced in the late 1960s. Secondary (mostly viscous) magnetizations were nearly all removed by alternating field demagnetization at 10 mT. We have used the so-called blanket cleaning method, generally at 10 mT. All sites were in basalt flows and gave good paleomagnetic results; none was rejected in toto, and only a few core specimens were magnetically unsatisfactory. Nearly all sites had eight independently oriented cores, and within-site angular standard deviations of directions range from 1° to 8°. We used both Fisher and Bingham statistics to analyze the data and found that many of the direction populations are strongly elongate along the paleomagnetic meridian, while the corresponding virtual pole (VGP) populations are essentially circularly distributed. The paleomagnetic poles, calculated as the means of VGPs, are as follows: Brunhes and Jaramillo, north latitude = 86.9°, east longitude = 245.1°, and 95% confidence radius A95 = 1.9°; Matuyama, latitude = 87.2°, longitude = 158.2°, and A95 = 3.8°; Gauss, latitude = 83.0°, longitude = 204.7°, and A95 = 7.0°. These
Chow, Sy-Miin; Lu, Zhaohua; Sherwood, Andrew; Zhu, Hongtu
2016-03-01
The past decade has evidenced the increased prevalence of irregularly spaced longitudinal data in social sciences. Clearly lacking, however, are modeling tools that allow researchers to fit dynamic models to irregularly spaced data, particularly data that show nonlinearity and heterogeneity in dynamical structures. We consider the issue of fitting multivariate nonlinear differential equation models with random effects and unknown initial conditions to irregularly spaced data. A stochastic approximation expectation-maximization algorithm is proposed and its performance is evaluated using a benchmark nonlinear dynamical systems model, namely, the Van der Pol oscillator equations. The empirical utility of the proposed technique is illustrated using a set of 24-h ambulatory cardiovascular data from 168 men and women. Pertinent methodological challenges and unresolved issues are discussed.
Leliaert, J.; Mulkers, J.; De Clercq, J.; Coene, A.; Dvornik, M.; Van Waeyenberge, B.
2017-12-01
Thermal fluctuations play an increasingly important role in micromagnetic research relevant for various biomedical and other technological applications. Until now, it was deemed necessary to use a time stepping algorithm with a fixed time step in order to perform micromagnetic simulations at nonzero temperatures. However, Berkov and Gorn have shown in [D. Berkov and N. Gorn, J. Phys.: Condens. Matter,14, L281, 2002] that the drift term which generally appears when solving stochastic differential equations can only influence the length of the magnetization. This quantity is however fixed in the case of the stochastic Landau-Lifshitz-Gilbert equation. In this paper, we exploit this fact to straightforwardly extend existing high order solvers with an adaptive time stepping algorithm. We implemented the presented methods in the freely available GPU-accelerated micromagnetic software package MuMax3 and used it to extensively validate the presented methods. Next to the advantage of having control over the error tolerance, we report a twenty fold speedup without a loss of accuracy, when using the presented methods as compared to the hereto best practice of using Heun's solver with a small fixed time step.
A Stochastic Continuous Time Model for Microgrid Energy Management
Heymann, Benjamin; Frédéric Bonnans, J; Silva, Francisco; Jimenez, Guillermo
2016-01-01
International audience; We propose a novel stochastic control formulation for the microgrid energy management problem and extend previous works on continuous time rolling horizon strategy to uncertain demand. We modelize the demand dynamics with a stochastic differential equation. We decompose this dynamics into three terms: an average drift, a time-dependent mean-reversion term and a Brownian noise. We use BOCOPHJB for the numerical simulations. This optimal control toolbox implements a semi...
International Nuclear Information System (INIS)
Colombino, A.; Mosiello, R.; Norelli, F.; Jorio, V.M.; Pacilio, N.
1975-01-01
A nuclear system kinetics is formulated according to a stochastic approach. The detailed probability balance equations are written for the probability of finding the mixed population of neutrons and detected neutrons, i.e. detectrons, at a given level for a given instant of time. Equations are integrated in search of a probability profile: a series of cases is analyzed through a progressive criterium. It tends to take into account an increasing number of physical processes within the chosen model. The most important contribution is that solutions interpret analytically experimental conditions of equilibrium (moise analysis) and non equilibrium (pulsed neutron measurements, source drop technique, start up procedures)
Delayed Stochastic Linear-Quadratic Control Problem and Related Applications
Directory of Open Access Journals (Sweden)
Li Chen
2012-01-01
stochastic differential equations (FBSDEs with Itô’s stochastic delay equations as forward equations and anticipated backward stochastic differential equations as backward equations. Especially, we present the optimal feedback regulator for the time delay system via a new type of Riccati equations and also apply to a population optimal control problem.
Ogawa, Shigeyoshi
2017-01-01
This book presents an elementary introduction to the theory of noncausal stochastic calculus that arises as a natural alternative to the standard theory of stochastic calculus founded in 1944 by Professor Kiyoshi Itô. As is generally known, Itô Calculus is essentially based on the "hypothesis of causality", asking random functions to be adapted to a natural filtration generated by Brownian motion or more generally by square integrable martingale. The intention in this book is to establish a stochastic calculus that is free from this "hypothesis of causality". To be more precise, a noncausal theory of stochastic calculus is developed in this book, based on the noncausal integral introduced by the author in 1979. After studying basic properties of the noncausal stochastic integral, various concrete problems of noncausal nature are considered, mostly concerning stochastic functional equations such as SDE, SIE, SPDE, and others, to show not only the necessity of such theory of noncausal stochastic calculus but ...