Averaging principle for slow-fast systems of rough differential equations via controlled paths

TL;DR

This paper establishes an averaging principle for slow-fast rough differential equations driven by independent rough paths, including fractional Brownian motion, using controlled path theory and time-discretization methods.

Contribution

It extends the averaging principle to rough differential systems with general rough paths, including fractional Brownian motion, within the controlled path framework.

Findings

Proved strong averaging principle for rough differential equations

Applicable to systems driven by fractional Brownian rough paths

Utilized Khas'minskii's time-discretizing method in rough path setting

Abstract

In this paper we prove the strong averaging principle for a slow-fast system of rough differential equations. The slow and the fast component of the system are driven by a rather general random rough path and Brownian rough path, respectively. These two driving noises are assumed to be independent. A prominent example of the driver of the slow component is fractional Brownian rough path with Hurst parameter between 1/3 and 1/2. We work in the framework of controlled path theory, which is one of the most widely-used frameworks in rough path theory. To prove our main theorem, we carry out Khas'minskii's time-discretizing method in this framework.