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This paper is devoted to studying the averaging principle for fast-slowsystem of rough differential equations driven by mixed fractional Brownianrough path. The fast component is driven by Brownian motion, while the slowcomponent is driven by fractional Brownian motion with Hurst index $H ~(1/3 <H\leq 1/2)$. Combining the fractional calculus approach to rough path theoryand Khasminskii's classical time discretization method, we prove that the slowcomponent strongly converges to the solution of the corresponding averagedequation in the $L^1$-sense. The averaging principle for a fast-slow system inthe framework of rough path theory seems new.

Averaging principle for fast-slow system driven by mixed fractional Brownian rough path