Averaging principles for mixed fast-slow systems driven by fractional Brownian motion

Abstract

We focus on fast-slow systems involving both fractional Brownian motion (fBm) and standard Brownian motion (Bm). The integral with respect to Bm is the standard Itô integral, and the integral with respect to fBm is a generalized Riemann–Stieltjes integral by means of fractional calculus. We establish an averaging principle in which the fast-varying diffusion process of the fast-slow systems acts as a “noise” to be averaged out in the limit. We show that the slow process has a limit in the mean square sense, which is characterized by the solution of stochastic differential equations driven by fBm whose coefficients are averaged with respect to the stationary measure of the fast-varying diffusion. An implication is that one can ignore the complex original systems and concentrate on the averaged systems instead. This averaging principle paves the way for reduction of computational complexity.

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