Linear-Quadratic Mean Field Stackelberg Games with State and Control Delays

Abstract

In this article, we consider a linear-quadratic mean field game between a leader (dominating player) and a group of followers (agents) under the Stackelberg game setting as proposed in [A. Bensoussan, M. Chau, and S. Yam, Appl. Math. Optim., 74 (2016), pp. 91-128], so that the evolution of each individual follower is now also subjected to delay effects from both his/her state and control variables, as well as those of the leader. The overall Stackelberg game is solved by tackling three subproblems hierarchically. Their resolution corresponds to the establishment of the existence and uniqueness of the solutions of three different forward-backward stochastic functional differential equations, which we manage by applying the unified continuation method as first developed in, for example, [Y. Hu and S. Peng, Probab. Theory Related Fields, 103 (1995), pp. 273-283] and [X. Xu, Fully Coupled Forward-Backward Stochastic Functional Differential Equations and Applications to Quadratic Optimal Control, preprint, arX...