Two-Person Zero-Sum Stochastic Linear-Quadratic Differential Games

TL;DR

This paper investigates stochastic linear-quadratic differential games, establishing conditions for saddle points and value finiteness, and providing Riccati equation solutions with examples illustrating key distinctions from deterministic cases.

Contribution

It introduces necessary and sufficient conditions for saddle points and value finiteness in stochastic LQ games, including Riccati equation solutions and their implications.

Findings

Finiteness of values does not guarantee saddle point existence in stochastic games.

A strongly regular Riccati solution ensures a saddle point in stochastic LQ games.

In deterministic cases, value finiteness and saddle point existence are equivalent.

Abstract

The paper studies the open-loop saddle point and the open-loop lower and upper values, as well as their relationship for two-person zero-sum stochastic linear-quadratic (LQ, for short) differential games with deterministic coefficients. It derives a necessary condition for the finiteness of the open-loop lower and upper values and a sufficient condition for the existence of an open-loop saddle point. It turns out that under the sufficient condition, a strongly regular solution to the associated Riccati equation uniquely exists, in terms of which a closed-loop representation is further established for the open-loop saddle point. Examples are presented to show that the finiteness of the open-loop lower and upper values does not ensure the existence of an open-loop saddle point in general. But for the classical deterministic LQ game, these two issues are equivalent and both imply the solvability of the Riccati equation, for which an explicit representation of the solution is obtained.