Linear-Quadratic Optimal Control Problems for Mean-Field Stochastic Differential Equations

Linear-quadratic optimal control problems are considered for mean-field stochastic differential equations with deterministic coefficients. By a variational method, the optimality system is derived, which is a linear mean-field forward-backward stochastic differential equation. Using a decoupling technique, two Riccati differential equations are obtained which are uniquely solvable under certain conditions. Then a feedback representation is obtained for the optimal control.