Applications of Stochastic Analysis and Control in Finance and Economics

University Of Southern California, Los Angeles CA

Investigators

Abstract

The research project will follow two main directions: (i). Principal-agent theory in continuous-time models with applications to the optimal compensation of company executives and fund managers. (ii). Numerical methods for high-dimensional optimization problems, Backward Stochastic Differential Equations (BSDEs) and degenerate Partial Differential Equations (PDEs), with applications in Finance and Economics. The planned research will make the theory of optimal contracts more compatible with the modern, more complex continuous-time models for financial markets by using 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 the time of the payoff is not necessarily fixed in advance. The second part of the project includes extending the work on finding efficient numerical methods for solving high-dimensional stochastic optimization problems, FBSDEs and degenerate PDEs, and to work on the existence/uniqueness theory for FBSDEs. The standard numerical PDE methods do not work in high-dimensions. Instead, these problems will be approached with a combination of dynamic programming, Malliavin Calculus, nonparametric regression methods, and existing numerical methods for FBSDEs. The first project is likely to have important applications to the question of how to compensate company executives and fund managers in an optimal way. The current practice for executive compensation is to grant options as a part of the compensation package, but there is a lot of recent discussion on whether this contributes to the manipulation of the company's stock by the executives, and also how these options should be accounted for in terms of the company's expenses. The research will address these issues and provide some answers in the framework of our mathematical models. In particular, anaysis will be conducted on whether some other forms of compensation, such as deferred options (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 the hardest practical problem in quantitative finance: to solve high-dimensional optimization problems. The most famous example of these is pricing high-dimensional American options. The techniques that are commonly used today are not appropriate for more complex and realistic models of security prices that are becoming a standard, due to the increased sophistication of market practitioners. Thus, it is important to explore new analytical and computational methods for this task.