Expansion of iterated Stratonovich stochastic integrals based on generalized multiple Fourier series

Abstract

The article is devoted to expansions of iterated Stratonovich stochastic integrals of multiplicities 1-4 on the base of the method of generalized multiple Fourier series. We prove the mean-square convergence of expansions in the case of Legendre polynomials as well as in the case of trigonometric functions. The considered expansions contain only one passage to the limit in contrast to its existing analogues. This property is very convenient for the mean-square approximation of iterated stochastic integrals. It is well-known that a prospective approach to numerical solving of Itô stochastic differential equations being adequate mathematical models of dynamical systems of various physical nature is one based on stochastic analogue of Taylor formula for the solutions to these equations. The iterated stochastic Stratonovich integrals are parts of so-called Taylor-Stratonovich expansion being one of the aforementioned stochastic analogues of Taylor formula. This is why the results of the paper can be applied to constructing strong numerical methods of convergence orders 1.0, 1.5 and 2.0 for Itô stochastic differential equations. The method of generalized multiple Fourier series does not require a partitioning of the integration interval for iterated stochastic Stratonovich integrals. This feature is essential since the mentioned integration interval is small playing a role of the integration step in numerical methods for Itô stochastic differential equations.