Optimization of the mean-square approximation procedures for iterated Stratonovich stochastic integrals of multiplicities 1 to 3 with respect to components of the multi-dimensional Wiener process based on Multiple Fourier–Legendre series

Abstract

The article is devoted to approximation of iterated Ito and Stratonovich stochastic integrals of multiplicities 1 to 3 by the method of multiple Fourier–Legendre series. The mentioned stochastic integrals are part of strong numerical methods with convergence order 1.5 for Ito stochastic differential equations with multidimensional noncommutative noise. These numerical methods are based on the so-called Taylor–Ito and Taylor–Stratonovich expansions. We calculate the exact lengths of sequences of independent standard Gaussian random variables required for the mean-square approximation of iterated Stratonovich stochastic integrals. Thus, the computational cost for the implementation of numerical methods can be significantly reduced.

On Traces of Linear Operators with Symmetrized Volterra-Type Kernels

“…For n 1 = n 2 , the required result follows from Lemma 1. For functions (12) and (13), conditions m + n = n 1 > n 2 = n and n = n 1 < n 2 = m + n, respectively, should be satisfied.…”

“…However, multiple (or iterated) stochastic integrals should be specified with respect to all possible combinations of components of the multidimensional Wiener process [1,3,12]. In [3,13,14], the trace convergence problem in this context is studied in detail with additional smoothness conditions on the weights ψ 1 , . .…”