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

Jingtao Shi,Xun Li,Jiongmin Yong

doi:10.1051/cocv/2021078

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.

Highlights

Let (Ω, F, P, F) be a complete filtered probability space, on which a one-dimensional standard Brownian motion W (·) is defined with F ≡ {Ft}t 0 being its natural filtration augmented by all the P-null sets in F, and E[ · ] denotes the expectation with respect to P
We look at closed-loop Nash equilibria for Problem (MF-SDG)
The following example shows that for the mean-field LQ non-zero sum stochastic differential game, it may happen that the closed-loop representations of open-loop Nash equilibria are different from the closed-loop Nash equilibria

Summary

Introduction

Let (Ω, F, P, F) be a complete filtered probability space, on which a one-dimensional standard Brownian motion W (·) is defined with F ≡ {Ft}t 0 being its natural filtration augmented by all the P-null sets in F , and E[ · ] denotes the expectation with respect to P. We consider two-person mean-field LQ non-zero sum stochastic differential games in an infinite horizon Both open-loop and closed-loop Nash equilibria are introduced. (ii) For MF-SDE LQ optimal control problems in an infinite horizon, we have established the equivalence among the solvability of a system of coupled algebraic Riccati equations, open-loop solvability, and closed-loop solvability This covers the relevant results found in Sun–Yong [46] where mean-field terms were absent.

Preliminaries

Mean-field LQ non-zero sum stochastic differential games

Open-loop Nash equalibria and their closed-loop representation

Closed-loop Nash equalibria and symmetric algebraic Riccati equations

Mean-field LQ zero-sum stochastic differential games

Examples

Concluding remarks