An Open-Loop Stackelberg Strategy for the Linear Quadratic Mean-Field Stochastic Differential Game

Abstract

This paper is concerned with the open-loop linear–quadratic (LQ) Stackelberg game of the mean-field stochastic systems in finite horizon. By means of two generalized differential Riccati equations, the follower first solves a mean-field stochastic LQ optimal control problem. Then, the leader turns to solve an optimization problem for a linear mean-field forward–backward stochastic differential equation. By introducing new state and costate variables, we present a sufficient condition for the existence and uniqueness of the Stackelberg strategy in terms of the solvability of some Riccati equations and a convexity condition. Furthermore, it is shown that the open-loop Stackelberg equilibrium admits a feedback representation involving the new state and its mean. Finally, two examples are given to show the effectiveness of the proposed results.