On non-zero-sum stochastic game problems with stopping times

This dissertation consists of three parts. We first study the continuous time non-zero-sum Dynkin game which is a multi-player non-cooperative game on stopping times. We show that the Dynkin game has a Nash equilibrium point for general stochastic processes. The study extends the result of Hamadene and Zhang. ❧ The second part is to study the value dynamics of Dynkin game. In a zero-sum Dynkin game, where one player’s cost is the other player’s benefit, the value process is characterized by a two-barrier reflected backward stochastic differential equation, see Cvitanic and Karatzas. We build a parallel result for non-zero-sum Dynkin game and propose a new form of equilibrium called time-autonomous, which is mainly used to overcome non-uniqueness of the equilibrium. Under this framework, we construct the equivalency relation between a reflected BSDE system with jumps and non-zero-sum Dynkin Game. ❧ Finally we study Principal-Agent problem on stopping times, which addresses time-inconsistent issue in the sense that Bellman’s principle does not hold. We propose a method to solve Principal-Agent problem in discrete time framework.