On Zero-sum Optimal Stopping Games

On a filtered probability space $(\mathrm{\Omega },\mathcal{F},P,\mathbb{F}=({\mathcal{F}}_t{)}_{t=0,\dots ,T})$, we consider stopper-stopper games $\overline V:=\inf_{\Rho\in\bT^{ii}}\sup_{\tau\in\T}\E[U(\Rho(\tau),\tau)]$and$\underline V:=\sup_{\Tau\in\bT^i}\inf_{\rho\in\T}\E[U(\Rho(\tau),\tau)]$ 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, $\text{T}$ is the set of stopping times, and ${\text{bT}}^i$ and ${\text{bT}}^{ii}$ are sets of mappings from $\text{T}$ to $\text{T}$ 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 $\mathrm{P}\in {\text{bT}}^{ii}$ and $\mathrm{T}\in {\text{bT}}^i$ for $\bar{V}$ and $\underline V$ respectively.