Applications of Stochastic Analysis and Control in Finance and Economics

点击翻译按钮获取中文摘要

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.