Averaging principle for distribution dependent stochastic differential equations driven by fractional Brownian motion and standard Brownian motion

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In this paper, we study distribution dependent stochastic differential equations driven simultaneously by fractional Brownian motion with Hurst index H>12 and standard Brownian motion. We first establish the existence and uniqueness theorem for solutions of the distribution dependent stochastic differential equations by utilising the Carathéodory approximation. Then under certain averaging condition, we show that the solutions of distribution dependent stochastic differential equations can be approximated by the solutions of the associated averaged distribution dependent stochastic differential equations in the sense of the mean square convergence.