Non-zero-sum stopping games in continuous time

Abstract

On a filtered probability space $(\Omega ,\mathcal{F},({\mathcal{F}}_t{)}_{t\in [0,\infty ]},\mathbb{P})$, we consider the two-player non-zero-sum stopping game $u^i:=\mathbb{E}[U^i(\rho ,\tau )],i=1,2$, where the first player choose a stopping strategy $\rho$ to maximize $u^1$ and the second player chose a stopping strategy $\tau$ to maximize $u^2$. Unlike the Dynkin game, here we assume that $U(s,t)$ is ${\mathcal{F}}_{s\vee t}$-measurable. Assuming the continuity of $U^i$ in $(s,t)$, we show that there exists an $\epsilon$-Nash equilibrium for any $\epsilon >0$.