  In this paper, we study a linear-quadratic optimal control problem for
mean-field stochastic differential equations driven by a Poisson random
martingale measure and a multidimensional Brownian motion. Firstly, the
existence and uniqueness of the optimal control is obtained by the classic
convex variation principle. Secondly, by the duality method, the optimality
system, also called the stochastic Hamilton system which turns out to be a
linear fully coupled mean-field forward-backward stochastic differential
equation with jumps, is derived to characterize the optimal control. Thirdly,
applying a decoupling technique, we establish the connection between two
Riccati equation and the stochastic Hamilton system and then prove the optimal
control has a state feedback representation.
