Stochastic averaging for the non-autonomous mixed stochastic differential equations with locally Lipschitz coefficients

TL;DR

This paper develops a stochastic averaging principle for non-autonomous slow-fast systems driven by mixed stochastic differential equations with locally Lipschitz coefficients, involving both Brownian and fractional Brownian motions, and proves convergence of the slow component to an averaged equation.

Contribution

It introduces a novel averaging method for mixed SDEs with locally Lipschitz coefficients driven by Bm and fBm, combining pathwise and Ito calculus techniques.

Findings

The slow component converges to the averaged equation in mean square sense.

The method handles both Brownian and fractional Brownian motions with Hurst parameter 1/2<H<1.

The approach extends averaging principles to more general stochastic systems.

Abstract

This paper investigates a non-autonomous slow-fast system, which is generalized by stochastic differential equations (SDEs) with locally Lipschitz coefficients, subjected to standard Brownian motion (Bm) and fractional Brownian motion (fBm) with Hurst parameter 1/2<H<1. We concentrate on how to handle both types of integrals with respect to Bm and fBm and the locally Lispchitz continuity. The pathwise approach and the Ito stochastic calculus are combined with the technique of stopping time to establish the averaging principle where the averaged equation is defined. Then, the slow component of the original slow-fast system converges to the solution of the proposed averaged equation in the mean square sense is verified.