Topics in Stochastic Control and Financial Mathematics

研究者研究随机控制和金融数学中的问题。第一个也是最重要的主题是用一种新版本的Perron方法对连续时间随机控制中的动态规划方法进行新的研究。该方法利用随机子解的上极值和随机超解的上极值，给出了两种粘滞解之间的挤压值函数。黏度解的唯一性（如果它成立）很容易表明该值函数是动态规划方程的唯一解。在不进行价值函数先验分析的情况下，得到了动态规划原理作为结论。这相当于在没有平滑性的情况下验证了粘度溶液的存在（类似于经典情况下的验证论点）。该项目的第二个主题在于通过建模基金经理的战略行为，理解高水位收费对基金经理的激励。研究者和他的同事研究基金经理在可用资产中的最优选择（导致基金股价），使得投资者的理性行为（对她来说是效用最大化）产生支付给基金经理的最大费用。第三个主题是理解最优停止的平均场博弈背景下的信息渗透的第一步。任何不确定性下的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，也适用于工程和生命科学。目前的项目主要包括一种新的技术方法来解决一类非常普遍的随机控制问题。新方法提供了对优化问题的更深层次的理解，并扩展了应用范围。此外，该项目还对基金经理的策略行为以及随机交互的人群之间的信息渗透进行了建模和研究。

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.