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

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

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A New Approach to the Series Expansion of Iterated Stratonovich Stochastic Integrals of Arbitrary Multiplicity with Respect to Components of the Multidimensional Wiener Process

Author(s):

Dmitriy Feliksovich Kuznetsov

Peter the Great Saint-Petersburg Polytechnic University
195251, Saint-Petersburg, Polytechnicheskaya ul., 29
Department of Higher Mathematics
Dr. Sc., Professor

sde_kuznetsov@inbox.ru

Abstract:

The article is devoted to a new approach to the series expansion of iterated Stratonovich stochastic integrals with respect to components of the multidimensional Wiener process. This approach is based on multiple Fourier-Legendre series as well as multiple trigonometric Fourier series. The theorem on the mean-square convergent expansion for the iterated Stratonovich stochastic integrals of arbitrary multiplicity is formulated and proved under the condition of convergence of trace series. This condition has been verified for integrals of multiplicities 1 to 5 and complete orthonormal systems of Legendre polynomials and trigonometric functions in Hilbert space. The Hu-Meyer formula and multiple Wiener stochastic integral were used in the proof of the mentioned theorem. The rate of mean-square convergence of the obtained expansions is found. The results of the article can be applied to the numerical integration of Ito stochastic differential equations with non-commutative noise in the framework of the approach based on the Taylor-Stratonovich expansion.

Keywords

References:

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