On a stopping game in continuous time

Abstract:

On a filtered probability space $(\Omega ,\mathcal {F},P,\mathbb {F}=(\mathcal {F}_t)_{0\leq t\leq T})$, we consider stopper-stopper games $\overline C:=\inf _{\boldsymbol {\rho }}\sup _{\tau \in \mathcal {T}} \mathbb {E}[U(\boldsymbol {\rho }(\tau ),\tau )]$ and $\underline C:=$ $\sup _{\boldsymbol {\tau }} \inf _{\rho \in \mathcal {T}}\mathbb {E}[U(\rho ,\boldsymbol {\tau } (\rho ))]$ in continuous time, where $U(s,t)$ is $\mathcal {F}_{s\vee t}$-measurable (this is the new feature of our stopping game), $\mathcal {T}$ is the set of stopping times, and $\boldsymbol {\rho },\boldsymbol {\tau }:\mathcal {T} \mapsto \mathcal {T}$ satisfy certain non-anticipativity conditions. We show that $\overline C=\underline C$, by converting these problems into a corresponding Dynkin game.