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In this paper, we study the optimal control problem for system driven bymixed fractional Brownian motion (including a fractional Brownian motion withHurst parameter $H>1/2$ and the underlying standard Brownian motion). By usingMalliavin calculus, we obtain the necessary condition the optimal controlshould satisfy. Through martingale representation theorem and the properties ofthe transforms operator, we give out the adjoint backward stochasticdifferential equation in a natural way. As a straightforward consequence, themaximum principle for control system driven by fractional Brownian motion andan independent Brownian motion is also deduced, which is different to theunderlying case. As an application, the linear quadratic case is investigatedto illustrate the main results.

Maximum Principle for Control System driven by Mixed Fractional Brownian Motion