Nonzero-Sum Games of Optimal Stopping for Markov Processes

Two players are observing a right-continuous and quasi-left-continuous strong Markov process X. We study the optimal stopping problem V-sigma(1) (x) = sup(tau) M-x(1) (tau, sigma sigma) for a given stopping time sigma (resp. V-tau(2)(x) = sup(sigma) M-x(2)(tau, sigma) for given tau) where M-x(1)(tau, sigma) = E-x[G(1)(X-tau)I(tau <= sigma) + H-1(X-sigma)1 (sigma < tau] with G(1), H-1 being continuous functions satisfying some mild integrability conditions (resp. M-x(2)(tau, sigma) = E-x[G(2)(X-sigma)1(sigma < tau) H-2(X-tau)I(tau <= sigma)] with G(2), H-2 being continuous functions satisfying some mild integrability conditions). We show that if sigma = sigma D-2 = inf{t >= 0 : X-t is an element of D-2} (resp. tau = tau(D1) = inf{t >= 0 : X-t is an element of D-1}) where D-2 (resp. D-1) has a regular boundary, then V sigma(1)(D2) (resp. V tau(2)(D1)) is finely continuous If D-2 (resp. D-1) is also (finely) closed then tau(sigma D2)(*) = inf{t >= 0 : X-t is an element of D-1(sigma D2)} (resp. sigma(tau D1)(*) = inf {t >= 0 : X-t is an element of D-2(tau D1)]) where D-1(sigma D2) = {V-sigma D2(1) = G(1)} (resp. D-2(tau D1) = {V-tau D1(2) = G(2)}) is optimal for player one (resp. player two). We then derive a partial superharmonic characterisation for V-sigma D2(1) (resp. V-tau D1(2)) which can be exploited in examples to construct a pair of first entry times that is a Nash equilibrium.