Stochastic averaging principle for dynamical systems with fractional Brownian motion

Abstract

Stochastic averaging for a class of stochastic differentialequations (SDEs) with fractional Brownian motion, of the Hurst parameter $H$ inthe interval $(\frac{1}{2},1)$, is investigated. An averaged SDE for theoriginal SDE is proposed, and their solutions are quantitatively compared. Itis shown that the solution of the averaged SDE converges to that of theoriginal SDE in the sense of mean square and also in probability. It is furtherdemonstrated that a similar averaging principle holds for SDEs understochastic integral of pathwise backward and forward types. Two examples arepresented and numerical simulations are carriedout to illustrate the averaging principle.