Stochastic Averaging Principle for Dynamical Systems with Fractional Brownian Motion

TL;DR

This paper investigates stochastic averaging for differential equations driven by fractional Brownian motion with Hurst parameter H in (1/2, 1), demonstrating convergence of solutions and validating the averaging principle through examples and simulations.

Contribution

It introduces an averaged SDE for systems with fractional Brownian motion and proves convergence of solutions, extending averaging principles to this class of stochastic systems.

Findings

Solution of averaged SDE converges to original in mean square

Solution converges in probability

Averaging principle applies to pathwise backward and forward SDEs

Abstract

Stochastic averaging for a class of stochastic differential equations (SDEs) with fractional Brownian motion, of the Hurst parameter H in the interval (1/2, 1), is investigated. An averaged SDE for the original SDE is proposed, and their solutions are quantitatively compared. It is shown that the solution of the averaged SDE converges to that of the original SDE in the sense of mean square and also in probability. It is further demonstrated that a similar averaging principle holds for SDEs under stochastic integral of pathwise backward and forward types. Two examples are presented and numerical simulations are carried out to illustrate the averaging principle.