New representations of the Hu-Meyer formulas and series expansion of iterated Stratonovich stochastic integrals with respect to components of a multidimensional Wiener process

Abstract

The article is devoted to the systematic derivation of new representations of the Hu-Meyer formulas. The formula expressing a multiple Wiener stochastic integral through the sum of multiple Stratonovich stochastic integrals and the formula expressing a multiple Stratonovich stochastic integral through the sum of multiple Wiener stochastic integrals are derived for the case of a multidimensional Wiener process. At that several different definitions of the multiple Stratonovich stochastic integral and several variants of sufficient conditions for the validity of the Hu-Meyer formulas are used. In particular, the proof method proposed by the author in 2006 is applied to obtain Hu-Meyers formulas based on generalized multiple Fourier series for the case of a multidimensional Wiener process. Of great importance for the numerical solution of Ito stochastic differential equations is the verification of sufficient conditions for the applicability of the Hu-Meyer formula (based on generalized multiple Fourier series) for the case of iterated Stratonovich stochastic integrals with respect to components of a multidimensional Wiener process. In the author's previous works, the indicated conditions were verified for iterated Stratonovich stochastic integrals of multiplicities 1 to 6 (the case of an arbitrary basis in the Hilbert space) and for iterated Stratonovich stochastic integrals of multiplicities 7 and 8 (the case of two special bases in Hilbert space (the trigonometric Fourier basis and the basis of Legendre polynomials)). Therefore, the results of the article will be usefull for constructing high-order strong numerical methods for non-commutative systems of Ito stochastic differential equations.