Mean-field linear-quadratic stochastic differential games in an infinite horizon^*

¹ Department of Applied Mathematics, Hong Kong Polytechnic University, Hong Kong, China.
² School of Mathematics, Shandong University, Jinan 250100, China.
³ Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA.

^** Corresponding author: shijingtao@sdu.edu.cn

Received: 13 July 2020
Accepted: 3 July 2021

Abstract

This paper is concerned with two-person mean-field linear-quadratic non-zero sum stochastic differential games in an infinite horizon. Both open-loop and closed-loop Nash equilibria are introduced. The existence of an open-loop Nash equilibrium is characterized by the solvability of a system of mean-field forward-backward stochastic differential equations in an infinite horizon and the convexity of the cost functionals, and the closed-loop representation of an open-loop Nash equilibrium is given through the solution to a system of two coupled non-symmetric algebraic Riccati equations. The existence of a closed-loop Nash equilibrium is characterized by the solvability of a system of two coupled symmetric algebraic Riccati equations. Two-person mean-field linear-quadratic zero-sum stochastic differential games in an infinite horizon are also considered. Both the existence of open-loop and closed-loop saddle points are characterized by the solvability of a system of two coupled generalized algebraic Riccati equations with static stabilizing solutions. Mean-field linear-quadratic stochastic optimal control problems in an infinite horizon are discussed as well, for which it is proved that the open-loop solvability and closed-loop solvability are equivalent.