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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

Автор(ы):

Dmitriy Feliksovich Kuznetsov

Peter the Great Saint-Petersburg Polytechnic University

sde_kuznetsov@inbox.ru

Аннотация:

The article is devoted to the development of a new approach to the series expansion of iterated Stratonovich stochastic integrals with respect to components of a multidimensional Wiener process. This 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 this work, expansions of iterated Stratonovich stochastic integrals of multiplicities 1 to 6 were obtained. At that, the expansions were constructed using two specific bases in Hilbert space. More precisely, Legendre polynomials and the trigonometric Fourier basis were used. In this paper, expansions of iterated Stratonovich stochastic integrals of multiplicities 1 to 4 are obtained on the base of arbitrary complete orthonormal systems of functions in Hilbert space. Sufficient conditions for the expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity are formulated in terms of trace series. The results of the article will be useful for construction of strong numerical methods with orders 1.0, 1.5 and 2.0 (based on the Taylor-Stratonovich expansion) for Ito stochastic differential equations with non-commutative noise.

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Ссылки:

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