On Zero-sum Optimal Stopping Games

Abstract

On a filtered probability space $(\Omega ,\mathcal{F},P,\mathbb{F}=({\mathcal{F}}_t{)}_{t=0,\dots ,T})$, we consider stopper-stopper games $\overline V:=\inf_{\Rho\in\bT^{ii}}\sup_{τ\in\T}\E[U(\Rho(τ),τ)]$ and $\underline V:=\sup_{\Tau\in\bT^i}\inf_{ρ\in\T}\E[U(\Rho(τ),τ)]$ in discrete time, where $U(s,t)$ is ${\mathcal{F}}_{s\vee t}$-measurable instead of ${\mathcal{F}}_{s\wedge t}$-measurable as is often assumed in the literature, $\T$ is the set of stopping times, and $\bT^i$ and $\bT^{ii}$ are sets of mappings from $\T$ to $\T$ satisfying certain non-anticipativity conditions. We convert the problems into a corresponding Dynkin game, and show that $\overline{V}=\underline{V}=V$, where $V$ is the value of the Dynkin game. We also get the optimal $\Rho\in\bT^{ii}$ and $\Tau\in\bT^i$ for $\overline{V}$ and $\underline{V}$ respectively.

Final version. To appear in Applied Mathematics and Optimization