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

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

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Multiple Ito and Stratonovich Stochastic Integrals: Fourier-legendre and Trigonometric Expansions, Approximations, Formulas

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:

In this book the problem of strong (mean-square) approximation of multiple Ito and Stratonovich stochastic integrals is systematically analyzed in the context of numerical integration of stochastic differential Ito equations. The presented monograph successfully uses the tool of multiple and iterative Fourier series, built in the space L2 and pointwise, for the strong approximation of multiple stochastic integrals and opens a new direction in researching 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 verified. We also obtained exact and approximate expressions for mean-square errors of approximation of multiple Ito stochastic integrals of multiplicity k. Significant practical material devoted to approximation of specific multiple Ito and Stratonovich stochastic integrals of 1-5 multiplicity using the system of Legendre polynomials and the system of trigonometric functions has been provided. The formulated in the book methods are compared with existing ones. We considered some weak approximations of multiple Ito stochastic integrals and proved the theorems about integration order replacement both for multiple Ito stochastic integrals and the multiple stochastic integrals according to martingales and martingale Poisson measures. Two families of analytical formulas for calculation of stochastic integrals were brought out. This book will be interesting for specialists dealing with the theory of stochastic processes, applied and computational mathematics as well as senior students and postgraduates of universities and technical institutes.

Keywords

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