Averaging principle for slow-fast fractional stochastic differential equations

We prove the averaging principle for a class of stochastic systems. The slow component is solution to a fractional differential equation, which is coupled with a fast component considered as solution to an ergodic stochastic differential equation driven by a standard Brownian motion. We establish the convergence of the slow component when the time-scale separation vanishes to the solution of the so-called averaged equation, which is an autonomous fractional differential equation, in the mean-square sense. Moreover, when the fast component does not depend on the slow component, we provide a rate of convergence depending on the order of the fractional derivative.