Applications of Stochastic Analysis and Control in Finance and Economics

该研究项目将遵循两个主要方向：（i）。连续时间模型中的委托代理理论及其在公司经理和基金经理最优报酬中的应用。 （ii）。高维优化问题的数值方法，倒向随机微分方程（BSDEs）和退化偏微分方程（PDEs），以及在金融和经济中的应用。 计划中的研究将通过使用几种不同的随机控制和优化方法，使最优合同理论与现代更复杂的金融市场连续时间模型更加兼容：偏微分方程方法，对偶/鞅方法，和正倒向随机微分方程方法。还计划将该理论扩展到以下情况，其中支付时间不一定提前固定。项目的第二部分包括扩展寻找有效数值方法的工作，以解决高三维随机优化问题、倒向随机微分方程和退化偏微分方程，并研究倒向随机微分方程的存在唯一性理论。标准的数值偏微分方程方法不适用于高维问题。相反，这些问题将与动态规划，Malliavin演算，非参数回归方法，和现有的数值方法FBSDES相结合。第一个项目可能对如何以最优方式补偿公司高管和基金经理的问题有重要的应用。目前的高管薪酬的做法是将期权作为薪酬方案的一部分，但最近有很多关于这是否有助于高管操纵公司股票的讨论，以及这些期权应该如何计入公司的费用。这项研究将解决这些问题，并在我们的数学模型框架内提供一些答案。特别是，将分析是否有其他形式的补偿，如延期期权（即将实施的几个大的美国公司），相对于现有的补偿形式的优势。第二个项目是有关的，也许是最困难的定量金融的实际问题：解决高维优化问题。其中最著名的例子是对高维美式期权的定价。由于市场从业者的日益成熟，目前普遍使用的技术并不适用于更复杂和更现实的证券价格模型，而这些模型正在成为一种标准。因此，为这项任务探索新的分析和计算方法是很重要的。

The research project will follow two main directions:(i). Principal-agent theory in continuous-time models withapplications to the optimal compensation of company executivesand fund managers. (ii). Numerical methods for high-dimensional optimization problems,Backward Stochastic Differential Equations (BSDEs) and degenerate Partial Differential Equations (PDEs), with applications inFinance and Economics. The planned research willmake the theory of optimal contracts more compatible with themodern, more complex continuous-time models for financial markets byusing several different methods of stochastic control and optimization:partial differential equations approach, duality/martingale approach,and the Forward-Backward Stochastic Differential Equation approach.It is also planned to extend the theory to the following cases in which thetime of the payoff is not necessarily fixed in advance.The second part of the project includes extending the work on findingefficient numerical methods for solving high-dimensional stochasticoptimization problems, FBSDEs and degenerate PDEs, and to work on the existence/uniqueness theory for FBSDEs. The standard numericalPDE methods do not work in high-dimensions. Instead, these problems willbe approached with a combination of dynamic programming, Malliavin Calculus, nonparametric regression methods, and existing numericalmethods for FBSDEs. The first project is likely to have important applications to the questionof how to compensate company executives and fund managers in an optimalway. The current practice for executive compensation is to grantoptions as a part of the compensation package, but there is a lot ofrecent discussion on whether this contributesto the manipulation of the company's stock by the executives, and also howthese options should be accounted for in terms of the company's expenses.The research will address these issues and provide some answers inthe framework of our mathematical models. In particular, anaysis will be conductedon whether some other forms of compensation, such as deferredoptions (as is about to be implemented by several large U.S. companies), have advantages relative to the existing forms of compensation.The second project is related to perhaps thehardest practical problem in quantitative finance: to solvehigh-dimensional optimization problems. The most famous example ofthese is pricing high-dimensional American options.The techniques that are commonly used today are not appropriate for morecomplex and realistic models of security prices that are becoming astandard, due to the increased sophistication of market practitioners.Thus, it is important to explore new analytical and computational methodsfor this task.