Русская версия
ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

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Stochastic Differential Equations: Theory and Practice of Numerical Solution. With Programs on MATLAB

Author(s):

Dmitriy Feliksovich Kuznetsov

Peter the Great Saint-Petersburg Polytechnic University
Russia, 195251, Saint-Petersburg, Polytechnicheskaya st., 29
Department of Higher Mathematics
Professor, Doctor of Physico-Mathematical Sciences

sde_kuznetsov@inbox.ru

Abstract:

This monograph is devoted to the problem of numerical integration of stochastic differential equations (SDE). The case of Ito SDE is systematically analized and the case of SDE with jump component is also considered. It is well known, that SDE are adequate mathematical models of dynamic systems under the influence of random disturbances. One of the effective approaches to numerical integration of Ito SDE is an approach based on Taylor-Ito and Taylor-Stratonovich expansions. In the presented book this approach is systematically analized. The book consist of 4 parts and 17 chapters. Let's deal with the content of this monograph according to chapters. In chapter 1 we gathered a support material which may be used while reading this book. We provided concepts of Markov processes, Ito and Stratonovich stochastic integrals, Ito formula, Ito SDE and SDE with jump component, stochastic integrals according to Poisson random measures and martingales. In chapter 2 we consider the mathematical models of dynamic systems of different physical nature under the influence of random disturbances on the base of SDE. Some mathematical problems for SDE also described in this chapter. We considered the problem of filtering (linear and non-linear), the problem of stochastic stability, the problem of stochastic control, the problem of estimation of parameters of stochastic systems and the probubilistic representations of solutions of Cauchy and Dirichlet problems for differential equations with partial derivatives or second order. Chapter 3 is devoted to some properties and formulas for stochastic integrals. We determined the class of multiple Ito stochastic integrals, for which with probability 1 the formulas of integration order replacement corresponding to the rules of classical integral calculus are reasonable. We proved the theorem of integration order replacement for the class of multiple Ito stochastic integrals. We analyzed many examples of this theorem usage. These results are generalized for the case of multiple stochastic integrals according to martingale. The formula of connection between multiple Ito and Stratonovich stochastic integrals of any fixed multiplicity k is proven. We brought out two families of analytical formulas for calculation of stochastic integrals on Ito processes of sufficiently general form. In chapter 4 we discuss the stochastic Taylor expansions. We consider the classical Taylor-Ito and Taylor-Stratonovich expansions and 4 new (so called) unified Taylor-Ito and Taylor-Stratonovich expansions. The most important feature of mentioned expansions is presence in them of multiple Ito or Stratonovich stochastic integrals, which play the key role for solving the problem of numerical integration of Ito SDE and SDE with jump component. The unified Taylor-Ito and Taylor-Stratonovich expansions are built on the base of theorem about integration order replacement in multiple Ito stochastic integrals (chapter 3). The unified Taylor-Ito and Taylor-Stratonovich expansions contains the minimal collections of multiple Ito and Stratonovich stochastic integrals, which can't be connected by linear relations. We called such collections as stochastic basises. Chapters 5 and 6 are devoted to strong (mean-square) approximations of collections of multiple Ito and Stratonovich stochastic integrals from the Taylor-Ito and Taylor-Stratonovich expansions. In chapters 5 we successfully use the tool of multiple and iterative Fourier series, built in the space L2 and pointwise, for the strong approximation of multiple Ito and Stratonovich stochastic integrals. We obtained a general result connected with expansion of multiple Ito stochastic integrals of any fixed multiplicity k, based on generalized multiple Fourier series converging in the space L2([t, T] x ... x [t, T]) (k-times). This result is adapted for multiple Stratonovich stochastic integrals of 1-4 multiplicity for Legendre polynomial system and system of trigonometric functions, as well as for other types of multiple stochastic integrals. The theorem on expansion of multiple Stratonovich stochastic integrals with any fixed multiplicity k, based on generalized iterative Fourier series converging pointwise is proven. In chapter 6 we obtained the expressions for mean-square errors of approximation of multiple Ito stochastic integrals in exact form for the case of multiplicity 1-5 and in general form for the case of any fixed multiplicity k. The evaluation for mean-square errors of approximation of multiple Ito stochastic integrals of any fixed multiplicity k is also obtained. We provided a significant practical material devoted to approximation of specific multiple Ito and Stratonovich stochastic integrals of 1-5 multiplicity from the Taylor-Ito and Taylor-Stratonovich expansions using the system of Legendre polynomials and the system of trigonometric functions. We compared the methods formulated in this book with existing methods. The last part of the chapter 6 is devoted to the weak approximations of multiple Ito stochastic integrals from the Taylor-Ito expansion. In chapters 7-9 we construct strong numerical methods for Ito SDE. Chapter 7 is devoted to explicit one-step strong numerical methods of orders of accuracy 0.5, 1.0, 1.5, 2.0, 2.5 and 3.0. The finite-difference modifications of Runge-Kutta type for some of mentioned methods are considered. The new step in this scientific field is connected with the use of methods of approximation of multiple Ito and Stratonovich stochastic integrals from the chapters 5 and 6 and with the use of unified Taylor-Ito and Taylor-Stratonovich expansions from cheapter 4. In chapter 8 we construct implicit one-step strong numerical methods for Ito SDE and in chapter 9 we construct explicit and implicit two-step and three-step strong numerical methods for Ito SDE of orders of accuracy 1.0, 1.5, 2.0 and 2.5. The Runge-Kutta type methods are also presented. In chapter 10 we consider weak numerical methods for Ito SDE. The most of them are well know, but we also constract several new numerical methods. In this chapter explicit, implicit, extrapolation and "predictor-corrector" methods as well as Runge-Kutta type methods are presented. Chapter 11 is devoted to numerical integration of linear stationary systems of Ito SDE. The numerical method, based on integral representation of solution of linear stationary system of Ito SDE and spectral expansion of diffusion matrix is considered. The order of accuracy of this method is investigated. At the second part of this chapter we consider another numerical methods for linear stationary systems of Ito SDE: method, based on approxmation of Wiener process by piecewise constant stochastic process and method, based on Taylor-Ito expansion and Legendre plynomials. In chapter 12 we consider the theory of numerical integration of SDE with jump component. We discuss the well known concept, which consider SDE with jump component as Ito SDE on the time intervals between the jumps of Poisson process. This approach allows to model separately of jump and diffusion components of the solution of SDE with jump component. For numerical modeling of diffusion component we can use the methods from the chapters 5-10. In chapter 13 we provide the library of MATLAB programs for numerical integration of linear stationary systems of Ito SDE, based on atgorithms from chapter 11. Numerical examples of use of this library is considered. In chapters 14-16 we demonstrate by numerical modeling the application of numerical methods, constructed in this monograph, to modeling of sample paths of solutions of systems of non-linear Ito SDE (chapter 14) and to numerical solution of mathematical problems by strong (chapter 15) and by weak (chapter 16) numerical methods. For the first time the numerical modeling of multiple Ito and Stratonovich stochastic integrals is realized with use of system of Legendre polynomials. Chapter 17 contains full texts of MATLAB programs, realizing the numerical experiments along the text of the book in whole.

Keywords

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