Expansion of Iterated Stratonovich Stochastic Integrals of Arbitrary Multiplicity Based on Generalized Iterated Fourier Series Converging Pointwise

Abstract

The article is devoted to the expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity $k$ $(k\in \mathbb{N})$ based on the generalized iterated Fourier series converging pointwise. The case of Fourier-Legendre series as well as the case of trigonometric Fourier series are considered in details. The obtained expansion provides a possibility to represent the iterated Stratonovich stochastic integral in the form of iterated series of products of standard Gaussian random variables. Convergence in the mean of degree $2n$ $(n\in \mathbb{N})$ of the expansion is proved. Some recent results on the expansion of iterated Stratonovich stochastic integrals of multiplicities 3 to 6 are given. The results of the article can be applied to the numerical solution of Ito stochastic differential equations.

80 pages. Minor changes in Sect. 5, 8, 9. arXiv admin note: text overlap with arXiv:1712.09746, arXiv:1712.09516