Research on Stochastic Optimization and Applications

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Research on Stochastic Optimization and ApplicationsHui WangProject AbstractMany stochastic optimization problems assume that the decision maker has the total freedom to intervene the system. That is, the control policies can be adjusted continuously and instantaneously, or the state processes can be stopped arbitrarily within some time interval. However, this assumption is often violated in practice. The first part of the project is concernedwith some new formulations for stochastic optimization problems in order to accommodate these practical constraints. These models share a common feature: random intervention times determined by exogenous signal processes. In the context of optimal control, the decision maker is allowed to adjust the level of control only at times when an exogenous process gives him a green light; for example, when an Poisson process makes a jump. In the area of optimal stopping, the state process is allowed to be stopped only at times when it receives certain signals from an exogenous process. Both formulations admit explicit solutions and can be applied to model practical constraints on control policies or stopping times. The second part of the project is concerned with a class of diffusion processes with jumps. The major effort is put on their applications to economics and finance. The reason for the introduction of such processes is that the classical diffusion model for stock prices cannot explain many empirical puzzles. For example, an abnormality called volatility smile is often observed in option pricing, and the return distributions of financial assets exhibit a leptokurtic feature. We intend to explain these phenomena, using the jump processes to model stock prices in a financial market or the values of economic projects in investment problems. The discontinuous behavior in the underlying processes requires new techniques to obtain explicit solutions to certain pricing and wealth optimization problems. The research project includes pricing commonly traded exotic options, analysis of utilitymaximization, and evaluation of investment projects.Stochastic optimization is one of the main topics in modern applied mathematics, with many applications in disciplines like engineering, biology, economics and finance, etc. The purpose of this research project is to systematically develop mathematical theories that are more faithful to real life problems, so that the conclusions drawn from them can be used with more confidence. For example, a better understanding for the uncertainty of stock prices can help agents reduce the risk in financial practice. However, in developing more realistic models, one must be aware of the subtle balance between complexity and mathematical tractability. A realistic yet too complicated model can be mathematically untractable. The goal of the project, therefore, is to develop analytically or numerically solvable models that embrace the essence of the practical problems. To this end, some new formulations are considered for general stochastic optimization problems in order to accommodate some practical constraints; for example, the controller can intervene the system only at times when an exogenous process sends out a certain signal. Also considered are some new models for economics and finance that incorporate the drastic changes of stock prices over short periods of time. These new models can be used to explain many empirical puzzles that the classical models fail to do, and to help agents develop better portfolios to reduce risk.