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

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

Approximation of Iterated Ito Stochastic Integrals of the Second Multiplicity Based on the Wiener Process Expansion Using Legendre Polynomials and Trigonometric Functions


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


The article is devoted to the mean-square approximation of iterated Ito stochastic integrals of the second multiplicity based on the Wiener process expansion using complete orthonormal systems of functions. The approximations of these stochastic integrals using Legendre polynomials and trigonometric functions are considered. In contrast to the method of expansion of iterated Ito stochastic integrals based on the Karhunen-Loeve expansion for the Wiener process, this method allows the use of different systems of basis functions, not only the trigonometric system of functions. The proposed method makes it possible to obtain expansions of stochastic integrals much easier than the methods based on generalized multiple Fourier series. The latter involve the calculation of coefficients of multiple Fourier series, which is a time-consuming task. The results of the article can be applied to the implementation of the Milstein method for the numerical integration of Ito stochastic differential equations and semilinear parabolic stochastic partial differential equations.



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