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

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

A Method of Expansion and Approximation of Repeated Stochastic Stratonovich Integrals Based On Multiple Fourier Series On Full Ortonormal Systems


D. F. Kuznetsov

Russia, 195251, St.-Petersburg, Polytechnicheskaja st. 29,
State Technical University,
Department of Higher mathematics,


We suggest a method of expansion and approximation of repeated stochastic Stratonovich integrals which makes possible to represent these integrals in the form of multiple series of products of standard independent Gaussian random values. Coefficients of these series are coefficients of multiple Fourier series on full ortonormal systems for special functions of several variables. A method considered in the paper allows to get a general formula for expansion and to evaluate an approximation error for repeated stochastic Stratonovich integrals of the arbitrary multiplicity k. The mean-square convergence of this method is proved. The comparison of the G.N.Milstein method and our method is given. Some examples of approximations and mean-square errors of approximations of repeated stochastic Stratonovich integrals are obtained. Our method is more general than the method of G.N.Milstein and can be applied to the numerical study of stochastic differential Ito equations, since, by this method and due to relations between Stratonovich and Ito stochastic integrals, we can approximate repeated stochastic Ito integrals from the Taylor-Ito expansion.

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