Linear‐Quadratic Optimal Control Problems for Mean‐Field Stochastic Differential Equations Driven by Fractional Brownian Motions

We study linear‐quadratic optimal control problems of mean‐field controlled stochastic differential systems driven by fractional Brownian motions with Hurst parameter with deterministic coefficients. By a variational method, we derive the optimality system, which is a mean‐field forward‐backward stochastic differential equation. When the diffusion term satisfies certain conditions, using a decoupling technique and Malliavin calculus, we obtain two Riccati differential equations that are uniquely solvable under certain conditions. Then we derive a feedback representation for the optimal control and the explicit solution of the mean‐field backward stochastic differential equation without Malliavin derivatives.