On Zero-Sum Optimal Stopping Games

Abstract

On a filtered probability space \((\Omega ,\mathcal {F},P,\mathbb {F}=(\mathcal {F}_t)_{t=0,\ldots ,T})\), we consider stopping games \(\overline{V}:=\inf _{{\varvec{\rho }}\in \mathbb {T}^{ii}}\sup _{\tau \in \mathcal {T}}\mathbb {E}[U({\varvec{\rho }}(\tau ),\tau )]\) and \(\underline{V}:=\sup _{{\varvec{\tau }}\in \mathbb {T}^i}\inf _{\rho \in \mathcal {T}}\mathbb {E}[U(\rho ,{\varvec{\tau }}(\rho ))]\) in discrete time, where U(s, t) is \(\mathcal {F}_{s\vee t}\)-measurable instead of \(\mathcal {F}_{s\wedge t}\)-measurable as is assumed in the literature on Dynkin games, \(\mathcal {T}\) is the set of stopping times, and \(\mathbb {T}^i\) and \(\mathbb {T}^{ii}\) are sets of mappings from \(\mathcal {T}\) to \(\mathcal {T}\) satisfying certain non-anticipativity conditions. We will see in an example that there is no room for stopping strategies in classical Dynkin games unlike the new stopping game we are introducing. We convert the problems into an alternative Dynkin game, and show that \(\overline{V}=\underline{V}=V\), where V is the value of the Dynkin game. We also get optimal \({\varvec{\rho }}\in \mathbb {T}^{ii}\) and \({\varvec{\tau }}\in \mathbb {T}^i\) for \(\overline{V}\) and \(\underline{V}\) respectively.