Non-zero-sum stopping games in continuous time

On a filtered probability space $(Ω,\mathcal{F}, (\mathcal{F}_t)_{t\in[0,\infty]}, \mathbb{P})$, we consider the two-player non-zero-sum stopping game $u^i := \mathbb{E}[U^i(ρ,τ)],\ i=1,2$, where the first player choose a stopping strategy $ρ$ to maximize $u^1$ and the second player chose a stopping strategy $τ$ 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 $ε$-Nash equilibrium for any $ε>0$.

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