Exact Calculation of the Mean-Square Error in the Method of Approximation of Iterated Ito Stochastic integrals, Based on Generalized Multiple Fourier Series

Abstract

The article is devoted to the developement of the method of expansion and mean-square approximation of iterated Ito stochastic integrals based on generalized multiple Fourier series converging in the sense of norm in the space $L_2([t, T]^k)$ ($k$ is the multiplicity of the iterated Ito stochastic integral). We obtain the exact and approximate expressions for the mean-square error of approximation of iterated Ito stochastic integrals of multiplicity $k$ ($k\in\mathbb{N}$) from the stochastic Taylor-Ito expansion in the framework of the mentioned method. As a result, we do not need to use redundant terms of expansions of iterated Ito stochastic integrals, that complicate the numerical methods for Ito stochastic differential equations. Moreover, we proved the convergence with propability 1 of the method of expansion of iterated Ito stochastic integrals based on generalized multiple Fourier series for the cases of multiple Fourier-Legendre series and multiple trigonometric Fourier series. Mean-square approximation of iterated Stratonovich stochastic integrals is also considered in the article. The results of the article can be applied to the high-order strong numerical methods for Ito stochastic differential equations as well as non-commutative semilinear stochastic partial differential equations with multiplicative trace class noise (in accordance with the mean-square criterion of convergence).