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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 with Respect to Components of a Multidimensional Wiener Process. the Case of Arbitrary Complete Orthonormal Systems in Hilbert Space. II

Author(s):

Dmitriy Feliksovich Kuznetsov

Doctor of Physical and Mathematical Sciences, Professor
of Department of Higher Mathematics,
Peter the Great Saint-Petersburg Polytechnic University

sde_kuznetsov@inbox.ru

Abstract:

The article is Part IV of the author's work devoted to a new approach to the series expansion of iterated Stratonovich stochastic integrals with respect to components of a multidimensional Wiener process. The above approach was proposed by the author in 2022 and is based on generalized multiple Fourier series in complete orthonormal systems of functions in Hilbert space. In the previous parts of the work, expansions of iterated Stratonovich stochastic integrals of multiplicities 1 to 6 (the case of Legendre polynomials and the trigonometric Fourier basis) and multiplicities 1 to 4 (the case of an arbitrary complete orthonormal system of functions in Hilbert space) were obtained. In this article, an expansion of iterated Stratonovich stochastic integrals of multiplicity 5 (the case of an arbitrary complete orthonormal system of functions in Hilbert space) is obtained. The mentioned expansion is generalized to the case of an arbitrary multiplicity of iterated Stratonovich stochastic integrals. The results of the article will be useful for construction of strong numerical methods with orders 1.0, 1.5, 2.0, ... (based on the Taylor-Stratonovich expansion) for systems of Ito stochastic differential equations with non-commutative noise.

Keywords

References:

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