Nonzero-sum stochastic differential games with stopping times and free boundary problemsc

One is given a diffusion process and two payoffs which depend on the process and on two stopping times tj, t2. Two players are to choose their respective stopping times t1? t2 so as to achieve a Nash equilibrium point. The problem whether such times exist is reduced to finding a “regular” solution (iq, of a quasi-variational inequality. Existence of a solution is established in the stationary case and, for one Space dimension, in the nonstationary case; for the latter situation, the solution is shown to be regular if the game is of zero sum. © 1977 American Mathematical Society.