Expectation of Stratonovich iterated integrals of Wiener processes

TL;DR

This paper derives a formula for the expectation of Stratonovich iterated integrals involving Wiener processes and time, providing a computational tool for stochastic differential equations.

Contribution

It introduces a new formula for expectations of Stratonovich iterated integrals with Wiener processes and offers a Mathematica implementation.

Findings

Derived a closed-form expectation formula for Stratonovich iterated integrals.

Provided a computational implementation in Mathematica.

Facilitates approximation of moments in stochastic differential equations.

Abstract

The solution of a (stochastic) differential equation (SDE) can be locally approximated by a stochastic expansion, a linear combination of iterated integrals. Quantities of interest, like moments, can then be approximated with the expansion. We present a formula for the case where the drivers of the equation are time and Wiener processes. We also present a Mathematica implementation of the result.