Averaging Principle for Fractional Stochastic Differential Equations Driven by Fractional Brownian Rough Paths

This paper investigates the averaging behavior of a class of slow–fast fractional stochastic differential equations that incorporate two distinct types of memory effects. The slow variable is governed by a Caputo fractional derivative of order α∈(0,1), and is driven by fractional Brownian motion with a Hurst index H∈(13,12], while the fast variable is driven by a standard Brownian motion. In this setting, the standard Ito^ calculus is no longer applicable. To overcome this difficulty, we reformulate the system within the framework of rough path theory, combined with Khasminskii’s time-discretization scheme. Under suitable regularity and mixing assumptions, we establish the existence and uniqueness of solutions for the coupled system. Furthermore, we prove that the slow component converges strongly to the solution of the associated averaged equation as the time-scale parameter tends to zero. Our result extends classical averaging principles to fractional slow–fast systems driven by rough stochastic signals, thereby providing a useful theoretical foundation for the analysis of memory-dependent stochastic models subject to irregular fractional Brownian perturbations.