On Zero-sum Optimal Stopping Games

TL;DR

This paper analyzes zero-sum stopping games with a novel measurability condition, converting them into Dynkin games and establishing the existence of a game value and optimal strategies.

Contribution

It introduces a new framework for stopping games with $_{sigvee t}$-measurable payoffs and proves the equivalence to Dynkin games, including optimal strategies.

Findings

The game value exists and is equal for both players.

The problems can be transformed into a Dynkin game framework.

Optimal strategies are characterized for both players.

Abstract

On a filtered probability space $(\Omega,\mathcal{F},P,\mathbb{F}=(\mathcal{F}_t)_{t=0,\dotso,T})$, 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, $\T$ 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.