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.本论文共包含三个研究部分。 首先，我们研究连续时间非零和邓金博弈（Dynkin game）——一类基于停时的多参与者非合作博弈。我们证明，针对一般随机过程，该邓金博弈存在纳什均衡点。本研究拓展了Hamadene与Zhang的研究成果。 ❧ 第二部分聚焦邓金博弈的价值动态研究。在零和邓金博弈中，一方参与者的成本即为另一方参与者的收益，其价值过程由双障碍反射倒向随机微分方程（reflected backward stochastic differential equation, BSDE）刻画，相关结论可见Cvitanic与Karatzas的研究。我们针对非零和邓金博弈构建了平行结论，并提出一种名为时间自治的新型均衡形式，该形式主要用于解决均衡的非唯一性问题。在此框架下，我们建立了带跳反射倒向随机微分方程系统与非零和邓金博弈之间的等价关系。 ❧ 最后，我们研究基于停时的委托-代理（Principal-Agent）问题，该问题针对贝尔曼原理不成立意义下的时间不一致性问题展开。我们提出一种可在离散时间框架下求解委托-代理问题的方法。