Nonzero-Sum Games of Optimal Stopping for Markov Processes

Abstract

Two players are observing a right-continuous and quasi-left-continuous strong Markov process X. We study the optimal stopping problem V^{1}_{sigma }(x)=sup _{tau } mathsf {M}_{x}^{1}(tau ,sigma ) for a given stopping time sigma (resp. V^{2}_{tau }(x)=sup _{sigma } mathsf {M}_{x}^{2}(tau ,sigma ) for given tau ) where mathsf {M}_{x}^{1}(tau ,sigma ) = mathsf {E}_{x} [G_{1}(X_{tau })I(tau le sigma ) + H_{1}(X_{sigma })I(sigma < tau )] with G_1,H_1 being continuous functions satisfying some mild integrability conditions (resp. mathsf {M}_{x}^{2}(tau ,sigma ) = mathsf {E}_{x} [G_{2}(X_{sigma })I(sigma < tau ) + H_{2}(X_{tau })I(tau le sigma )] with G_2,H_2 being continuous functions satisfying some mild integrability conditions). We show that if sigma = sigma _{D_{2}} = inf {t ge 0: X_t in D_2} (resp. tau = tau _{D_{1}} = inf {t ge 0: X_t in D_1}) where D_{2} (resp. D_1) has a regular boundary, then V^{1}_{sigma _{D_{2}}} (resp. V^{2}_{tau _{D_{1}}}) is finely continuous. If D_{2} (resp. D_1) is also (finely) closed then tau _*^{sigma _{D_2}} = inf {t ge 0: X_{t} in D_{1}^{sigma _{D_{2}}}} (resp. sigma _{*}^{tau _{D_1}} = inf {t ge 0: X_{t} in D_{2}^{tau _{D_{1}}}}) where D_{1}^{sigma _{D_{2}}} = {V^{1}_{sigma _{D_{2}}} = G_{1}} (resp. D_{2}^{tau _{D_{1}}} = {V^{2}_{tau _{D_{1}}} = G_{2}}) is optimal for player one (resp. player two). We then derive a partial superharmonic characterisation for V^{1}_{sigma _{D_2}} (resp. V^{2}_{tau _{D_1}}) which can be exploited in examples to construct a pair of first entry times that is a Nash equilibrium.