Closed-Loop Solvability of Linear Quadratic Mean-Field Type Stackelberg Stochastic Differential Games

This paper is devoted to a Stackelberg stochastic differential game for a linear mean-field type stochastic differential system with a mean-field type quadratic cost functional over a finite horizon. Coefficients in the state equation and weighting matrices in the cost functional are all deterministic. Closed-loop Stackelberg equilibrium strategies are introduced that are independent of initial states. It begins by solving the follower’s stochastic linear quadratic optimal control problem. By transforming the original problem into a new one with a known optimal control, the closed-loop optimal strategy of the follower is characterized by two coupled Riccati equations and a linear mean-field type backward stochastic differential equation. Then the leader turns to solve a stochastic linear quadratic optimal control problem for a mean-field type forward-backward stochastic differential equation. Necessary conditions for the existence of closed-loop optimal strategies for the leader are given by the existence of two coupled Riccati equations with a linear mean-field type backward stochastic differential equation. The solvability of Riccati equations of the leader’s problem is discussed, particularly in cases where the diffusion term of the state equation does not contain the control process of the follower. Moreover, the leader’s value function is expressed via two backward stochastic differential equations and two Lyapunov equations. Finally, a numerical example is given to show the effectiveness of the proposed results.