A closed-loop saddle point for zero-sum linear-quadratic stochastic differential games with mean-field type

Abstract

This paper is concerned with a linear-quadratic zero-sum differential game with mean-field type. The notions of explicit and implicit strategy laws are proposed. Based on them, a closed-loop formulation for saddle points in the mixed-strategy-law form is established. In order to construct the saddle point, the classical entire completion-of-square argument is developed to a four-step-completion-of-square argument, and a pair of optimality conditions is proposed.