Optimal Stopping Games for Markov Processes

Let X=(Xₜ)t ≥ 0 be a strong Markov process, and let G₁,\, G₂, and G₃ be continuous functions satisfying G₁ ≤ G₃ ≤ G₂ and Eₓₜ Gᵢ(Xₜ) < ∞ for $i=1,2,3$. Consider the optimal stopping game where the sup-player chooses a stopping time τ to maximize, and the inf-player chooses a stopping time σ to minimize, the expected payoff Mₓ(τ,σ) = Eₓ [G₁(X_τ)\;\! I(τ\! <\! σ) + G₂(X_σ)\;\! I(σ\! <\! τ) + G₃(X_τ)\;\! I(τ\! =\! σ)], where X₀=x under P\!x. Define the upper value and the lower value of the game by V^*(x) = inf_σ _τ Mₓ(τ,σ)~ and~ V_*(x) = _τ inf_σ Mₓ(τ,σ), respectively, where the horizon T (the upper bound for τ and σ above) may be either finite or infinite (it is assumed that G₁(XT)=G₂(XT) if T is finite and t → ∞ G₂(Xₜ) ≤ t → ∞ G₁(Xₜ) if T is infinite). If X is right-continuous, then the Stackelberg equilibrium holds, in the sense that V^*(x)=V_*(x) for all x with V:=V^*=V_* defining a measurable function. If X is right-continuous and left-continuous over stopping times (quasi-left-continuous), then the Nash equilibrium holds, in the sense that there exist stopping times τ_* and σ_* such that Mₓ(τ,σ_*) ≤ Mₓ(τ_*,σ_*) ≤ Mₓ(τ_*,σ) for all stopping times τ and σ, implying also that V(x)=Mₓ(τ_*,σ_*) for all x. Further properties of the value function V and the optimal stopping times τ_* and σ_* are exhibited in the proof.