Stochastic averaging principle for dynamical systems with fractional Brownian motion

Stochastic averaging for a class of stochastic differential equations (SDEs) with fractional Brownian motion, of the Hurst parameter H in the interval (12, 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 path-wise backward and forward types. Two examples are presented and numerical simulations are carried out to illustrate the averaging principle.