Application of Walsh series to represent iterated Stratonovich stochastic integrals

Abstract

The goal of the paper is to represent iterated Stratonovich stochastic integrals using orthonormal expansions of functions by the Walsh series, to construct an algorithm for modeling iterated Stratonovich stochastic integrals, and to test this algorithm by numerical experiments. Obtained results may be used in the constructing high-order numerical methods based on the Taylor –Stratonovich expansion and the Taylor– It^o expansion. Therefore, they can be applied in a lot of problems of physics, engineering, economics and finance, when it is necessary to solve numerically stochastic differential equations.

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.…”

“…In fact, we can only study functions (1) for this purpose [3]. 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 , .…”

Spectral Representations of Iterated Stochastic Integrals and Their Application for Modeling Nonlinear Stochastic Dynamics

“…For iterated Itô stochastic integrals, it avoids relations with the indicators [20] or the Wick product [31] of random variables that correspond to Wiener processes. In this context, iterated Itô and Stratonovich stochastic integrals are considered earlier with the simplest weight functions [32][33][34]. Here, we propose a general case of weight functions and consider mixed-type iterated stochastic integrals of arbitrary multiplicity.…”