On a stopping game in continuous time

Abstract

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