CAREER: Topics in Optimal Stopping and Control

该项目的第一部分将考虑最优停止问题，其中最优停止理论的众所周知的结果不适用于该问题。一个例子是当套利机会很弱或存在股价泡沫时的美式期权定价问题。这一部分的问题是关于确定所谓柯西问题解唯一的充要条件。分位数套期保值(高成功概率套期保值)问题，其解的特征是非线性偏微分方程组，也将在上述背景下进行研究。在第二部分中，我将为最优停止问题发展一个新的理论，当统计期望算子被衡量未来回报的替代方法所取代时，例如通过所谓的风险衡量。这些问题与控制和停止的随机微分对策有关。这类游戏的鞍点将被确定。在某些特殊情况下，它们是与拟线性(积分)偏微分方程组相关的自由边界。我们将研究值函数和自由边界曲线的规律性。本文还将分析风险厌恶与自由边界形状之间的关系。在第三部分中，我们将使用路径比较和凸对偶方法来求解以最大化达到某一目标的概率为目标的优化问题。考虑到大型投资者面临监管风险约束的当前经济环境，本部分将考虑的带有风险约束的效用最大化问题是非常相关的。该项目将导致最优停止和随机控制理论的几个方法论/理论的发展。作为副产品，这些发展将帮助我们理解线性/非线性偏微分方程解的存在性、唯一性和正则性问题。我们的结果将解决数学金融中重要的定价和套期保值问题。该项目补充了风险度量理论的最新发展，将风险度量作为优化标准来解决决策问题，这是管理金融风险的重要一步。

The first part of the project will consider optimal stopping problems in which the well-known results of the optimal stopping theory do not apply. An example is the American option pricing problem when there are weak arbitrage opportunities or a stock price bubble. The problems in this part are related to determining necessary and sufficient conditions for the uniqueness of solutions of the so-called Cauchy problems. Quantile hedging (hedging with high probability of success) problems, solutions of which are characterized by non-linear partial differential equations, will also be investigated in the above context. In the second part, I will develop a new theory for optimal stopping problems when the statistical expectation operator is replaced by alternative ways of measuring future rewards, for example by the so-called risk measures. These problems are related to stochastic differential games of control and stopping. Saddle points of such games will be determined. In some special cases, these are related free boundaries with quasi-linear (integro) partial differential equations. Regularity of the value function and the free boundary curve will be investigated. The relationship between the risk aversion and the shape of the free boundary will also be analyzed. In the third part, pathwise comparison and convex duality methods will be used to solve optimization problems with objectives of maximizing the probability of reaching certain goals. Utility maximization problems with risk constraints that will be considered in this part are quite relevant given the current economic environment in which large investors face regulatory risk constraints.The project will lead to several methodological/theoretical developments in the theory of optimal stopping and stochastic control. As a by-product, these developments will help us understand the existence, uniqueness, and regularity questions in linear/non-linear partial differential equations. Our results will resolve important pricing and hedging questions in Mathematical Finance. This project complements the recent developments in the theory of risk measures by addressing decision making problems using these measures as optimization criteria, which is an important step in managing financial risk.