Optimal stopping games for Markov processes

Let X = (X-t)(t >= 0) be a strong Markov process, and let G(1), G(2), and G(3) be continuous functions satisfying G(1) <= G(3) <= G(2) and E-x sup(t) \G(i)(X-t)\ < infinity for i = 1, 2, 3. Consider the optimal stopping game where the sup-player chooses a stopping time T to maximize, and the inf-player chooses a stopping time sigma to minimize, the expected payoff M-x(tau, sigma) = E-x[G(1)(X-tau)I(tau < sigma) + G(2)(X-sigma)I(sigma < tau) + G(3)(X-tau)I(tau = sigma)], where X-0 = x under P-x. Define the upper value and the lower value of the game by V *(x) = inf(sigma) sup(tau) M-x(tau, sigma) and V-*(x) = sup(tau) infs M-x(tau,sigma), respectively, where the horizon T ( the upper bound for tau and sigma above) may be either finite or infinite (it is assumed that G(1)(X-T) = G(2)(X-T) if T is finite and lim inf(t-->infinity) G(2)(X-t) <= lim sup tau-->8 G(1)(X-t) 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* de. ning 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 tau(*) and sigma(*) such that M-x(tau, sigma(*)) = M-x(tau(*),sigma(*)) = M-x(tau(*),sigma) for all stopping times t and s, implying also that V (x) = M-x(tau(*),sigma(*)) for all x. Further properties of the value function V and the optimal stopping times tau(*) and sigma(*) are exhibited in the proof.