带学习的稳健最优停止问题研究

最优停止问题在经济和金融等领域有广泛的应用。当决策者面对奈特不确定性时，通常需要考虑稳健的最优停止策略。本项目研究当决策者能够通过观测去学习模型的参数时，如何建立稳健的最优停止准则。这个问题与一类反射倒向随机微分方程密切相关。主要研究内容包括：1、基于观测到的信息，构造出刻画决策者不确定性的先验概率测度集合，并在弱框架下建立决策者的随机微分递归效用方程。2、在最大最小（maxmin）递归效用框架下推导出值函数满足的变分不等式，并证明它是一个自由边界的微分方程（偏微分方程）的解，从而推导出稳健的最优停止准则。3、建立带学习的稳健最优停时问题与对应的反射倒向随机微分方程之间的联系，并在弱框架下证明推导出的反射倒向随机微分方程解的存在唯一性。4、建立离散时间下带学习的稳健最优停止问题的模型；5、研究带学习的反射倒向随机微分方程的蒙特卡洛数值算法，并计算美式期权定价等金融中的资产定价问题。

The optimal stopping problem has a wide range of applications in the fields of economy and finance. When decision makers face Knight uncertainty, they usually need to consider a robust optimal stopping strategy. This project examines how to establish a robust optimal stopping criterion when decision makers can observe the parameters of the learning model. This problem is closely related to one kind of reflected backward stochastic differential equation. The main research contents include: 1. Based on the observed information, construct a set of prior probability measures that characterize the uncertainty of decision makers, and establish a stochastic differential recursive utility for decision makers in a weak formulation. 2. In the Maxmin recursive utility framework, the variational inequality satisfied by the value function is derived, and it is proved that it is a solution of a free boundary differential equation (partial differential equation), thus deducing a robust optimal stopping criterion. 3. Establish the relationship between the robust optimal stopping problem with learning and the corresponding reflected backward stochastic differential equation, and prove the existence and uniqueness of the solution of the backward stochastic differential equation under the weak framework. 4. Establish a model of robust optimal stopping problem with learning under discrete time; 5. Study Monte Carlo numerical algorithm with reflected backward stochastic differential equations with learning, and calculate asset pricing problems in financials such as American option pricing.