Mean-Field Linear-Quadratic Stochastic Differential Games

TL;DR

This paper investigates the conditions for the existence of saddle points in two-player zero-sum mean-field linear-quadratic stochastic differential games, providing Riccati equation solutions and feedback representations.

Contribution

It establishes necessary and sufficient conditions for open-loop saddle points using Riccati equations and introduces an approximation method for cases satisfying only the necessary condition.

Findings

Unique solutions to Riccati equations under sufficient conditions

Representation of saddle points as linear feedback controls

Convergence of approximate sequences to open-loop saddle points

Abstract

The paper is concerned with two-person zero-sum mean-field linear-quadratic stochastic differential games over finite horizons. By a Hilbert space method, a necessary condition and a sufficient condition are derived for the existence of an open-loop saddle point. It is shown that under the sufficient condition, the associated two Riccati equations admit unique strongly regular solutions, in terms of which the open-loop saddle point can be represented as a linear feedback of the current state. When the game only satisfies the necessary condition, an approximate sequence is constructed by solving a family of Riccati equations and closed-loop systems.The convergence of the approximate sequence turns out to be equivalent to the open-loop solvability of the game, and the limit is exactly an open-loop saddle point, provided that the game is open-loop solvable.