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

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.