Almost Sure Averaging for Fast-slow Stochastic Differential Equations via Controlled Rough Path

TL;DR

This paper develops an almost sure averaging method for coupled stochastic differential equations driven by fractional Brownian motion with low regularity, using controlled rough path techniques and random dynamical systems.

Contribution

It introduces a novel averaging approach for slow-fast SDEs with fractional Brownian motion, proving almost sure convergence via controlled rough paths and RDS.

Findings

Solution of slow component converges almost surely to the averaged equation

Employs controlled rough path approach for low-regularity FBM

Uses RDS to find stationary solutions with exponential attraction

Abstract

This paper establishes the averaging method to a coupled system consisting of two stochastic differential equations which has a slow component driven by fractional Brownian motion (FBM) with less regularity $1/3< H \leq 1/2$ and a fast dynamics under additive FBM with Hurst-index $1/3< \hat H \leq 1/2$. We prove that the solution of the slow component converges almost surely to the solution of the corresponding averaged equation using the approach of time discretization and controlled rough path. To do this, we employ the random dynamical system (RDS) to obtain a stationary solution by an exponentially attracting random fixed point of the RDS generated by the non-Markovian fast component.