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

Abstract:

One is given a diffusion process and two payoffs which depend on the process and on two stopping times ${\tau _1},{\tau _2}$. Two players are to choose their respective stopping times ${\tau _1},{\tau _2}$ so as to achieve a Nash equilibrium point. The problem whether such times exist is reduced to finding a “regular” solution $({u_1},{u_2})$ 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.