Topics in Stochastic Control and Financial Mathematics

随机控制和金融数学专题

• 批准号: 1211988
• 负责人: Mihai Sirbu
• 金额: $ 29.19万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-09-15 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1211988&HistoricalAwards=false
• 关键词: Topics; Stochastic; Control; Financial; Mathematics

SirbuDMS-1211988 The investigator studies problems in stochastic control and financial mathematics. The first and most important topic is a new look at the dynamic programming method in continuous-time stochastic control using a novel version of Perron's method. Taking the supremum of stochastic sub-solutions and infimum of stochastic super-solutions, the new method provides two viscosity solutions squeezing between them the value function. Uniqueness of the viscosity solution (in case it holds) then easily shows that the value function is the unique solution of the dynamic programming equation. The dynamic programming principle is obtained as a conclusion using this approach, without any a priori analysis of the value function. This amounts to verification without smoothness of the existence of a viscosity solution (similar to the verification argument in the classic case). The second topic of the project resides in understanding the incentives of high-watermark fees on the fund manager, by modeling his/her strategic behavior. The investigator and his colleagues study the optimal choice of the fund manager among available assets (that leads to the fund share price), such that the rational behavior of the investor (utility maximization on her side) yields maximal fees paid to the manager. The third topic is a first step into understanding information percolation in the context of mean-field games of optimal stopping. Any decision under uncertainty can be modeled as a stochastic control/optimization problem. This applies not only to finance and economics but to engineering and life sciences. The current project mainly consists in a new technical approach to a very general class of stochastic control problems. The new approach provides a deeper understanding of the optimization problems, and it also extends the scope of applications. In addition, the project models and studies the strategic behavior of fund managers, as well as the percolation of information among populations that interact randomly.

Sirbudms-1211988研究者研究了随机控制和金融数学方面的问题。 第一个也是最重要的话题是使用Perron方法的新颖版本的连续时间随机控制中的动态编程方法的新介绍。 新方法以随机子分解和随机超溶液的最高效果，提供了两种粘度解决方案，它们之间挤压了它们的值函数。 然后，粘度解的唯一性（如果粘度解决方案持有），然后很容易地表明该值函数是动态编程方程的唯一解决方案。 使用这种方法获得了动态编程原理作为结论，而无需对价值函数进行任何先验分析。 这相当于验证，而没有粘度解决方案的平滑度（类似于经典情况下的验证参数）。 该项目的第二个主题是通过对其战略行为进行建模，了解基金经理的高水标费用的激励措施。 调查员及其同事研究了可用资产中基金经理的最佳选择（这导致了基金股价），因此投资者的合理行为（公用事业最大化在她这方面）可以使经理支付的最高费用。 第三个主题是在最佳停止均值游戏中了解信息渗透的第一步。 不确定性下的任何决策都可以建模为随机控制/优化问题。 这不仅适用于金融和经济学，还适用于工程和生命科学。 当前的项目主要是针对非常一般的随机控制问题的新技术方法。 新方法提供了对优化问题的更深入的了解，并且还扩展了应用程序的范围。 此外，项目模型和研究基金经理的战略行为，以及随机相互作用的人群之间信息的渗透。