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

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