Long-Time Behavior of Zero-Sum Linear-Quadratic Stochastic Differential Games

TL;DR

This paper studies the long-term behavior of zero-sum linear-quadratic stochastic differential games, showing that strategies and states tend to stable distributions over time and solving related infinite horizon problems.

Contribution

It demonstrates the exponential turnpike property for strategies and states, and provides the first closed-loop solutions for the infinite horizon differential game.

Findings

Strategies and states stay near invariant distributions for most of the time horizon.

Established the exponential turnpike property under certain conditions.

Derived closed-loop representations for the infinite horizon saddle strategy.

Abstract

The paper investigates the long-time behavior of zero-sum linear-quadratic stochastic differential games, aiming to demonstrate that, under appropriate conditions, both the saddle strategy and the optimal state process exhibit the exponential turnpike property. Namely, for the majority of the time horizon, the distributions of the saddle strategy and the optimal state process closely stay near certain (time-invariant) distributions $\nu_1^*$, $\nu_2^*$ and $\mu^*$, respectively. Additionally, as a byproduct, we solve the infinite horizon version of the differential game and derive closed-loop representations for its open-loop saddle strategy, which has not been proved in the literature.