Linear Quadratic Stochastic Two-Person Zero-Sum Differential Games in an Infinite Horizon

TL;DR

This paper studies infinite horizon linear quadratic stochastic two-player zero-sum differential games, characterizing saddle points via algebraic Riccati equations and establishing key solvability results for associated backward stochastic differential equations.

Contribution

It introduces conditions for the existence of closed-loop saddle points and links them to algebraic Riccati equations in an infinite horizon setting.

Findings

Existence of closed-loop saddle points characterized by algebraic Riccati equations.

Unique solvability of a class of linear backward stochastic differential equations.

Framework applicable to stochastic differential games with constant coefficients.

Abstract

This paper is concerned with a linear quadratic stochastic two-person zero-sum differential game with constant coefficients in an infinite time horizon. Open-loop and closed-loop saddle points are introduced. The existence of closed-loop saddle points is characterized by the solvability of an algebraic Riccati equation with a certain stabilizing condition. A crucial result makes our approach work is the unique solvability of a class of linear backward stochastic differential equations in an infinite horizon.