CAREER: Topics in Optimal Stopping and Control

职业：最佳停止和控制主题

• 批准号: 0955463
• 负责人: Erhan Bayraktar
• 金额: $ 40万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-09-01 至 2016-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0955463&HistoricalAwards=false
• 关键词: CAREER; Topics; Optimal; Stopping; Control

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.

该项目的第一部分将考虑最佳停止问题，其中最佳停止理论的众所周知的结果不适用。一个例子是当套利机会较弱或股票价格泡沫时，美式期权的定价问题。这部分的问题涉及确定所谓柯西问题解唯一性的充要条件。分位数套期保值（高成功概率的套期保值）问题，其解决方案以非线性偏微分方程为特征，也将在上述背景下进行研究。在第二部分中，当统计期望算子被衡量未来回报的替代方法（例如所谓的风险度量）取代时，我将开发一种关于最优停止问题的新理论。 这些问题与控制和停止的随机微分博弈有关。此类游戏的鞍点将被确定。 在某些特殊情况下，这些是与准线性（积分）偏微分方程相关的自由边界。 将研究价值函数和自由边界曲线的规律性。还将分析风险厌恶程度与自由边界形状之间的关系。 在第三部分中，将使用路径比较和凸对偶方法来解决以最大化达到特定目标的概率为目标的优化问题。考虑到当前大投资者面临监管风险约束的经济环境，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一些方法/理论发展。作为副产品，这些发展将帮助我们理解线性/非线性偏微分方程中的存在性、唯一性和规律性问题。我们的结果将解决数学金融中重要的定价和对冲问题。该项目通过使用这些措施作为优化标准来解决决策问题，补充了风险措施理论的最新发展，这是管理金融风险的重要一步。