Mean-Field Linear-Quadratic Stochastic Differential Games in an Infinite Horizon

TL;DR

This paper studies mean-field linear-quadratic stochastic differential games on an infinite horizon, providing conditions for the existence of open-loop and closed-loop Nash equilibria and saddle points through Riccati equations.

Contribution

It introduces new characterizations for Nash equilibria and saddle points in mean-field LQ stochastic games using coupled Riccati equations, extending existing theory to infinite horizon cases.

Findings

Existence of open-loop Nash equilibrium characterized by mean-field forward-backward SDEs.

Closed-loop Nash equilibrium represented via coupled algebraic Riccati equations.

Existence of saddle points in zero-sum games characterized by generalized algebraic Riccati equations.

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. 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 time 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.