Problems in Stochastic Control, Incomplete Markets, and Stochastic Limit Theorems

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The proposed project consists of four parts: (1) Solving the optimal stopping problems associated with several quickest detection problems for processes that jump. The goal is to provide novel optimization techniques for such processes and study properties of the corresponding variational inequalities. (2) Developing stochastic control techniques to study the problem of how an individual (retiree) should invest her wealth in a risky financial market in order to minimize the probability that she outlives her wealth, minimizes her life-time shortfall. In this case, the aim is to develop the earlier results of L-infinity control and to analyze the associated variational equalities for more realistic market models. (3) Developing new pricing principles for incomplete markets, with the objective of providing new insights into pricing and hedging derivative securities in incomplete markets. This part also includes developing solutions of impulse and singular control problems for any one-dimensional diffusion (with decision making delay). (4) Developing stochastic limit theorems for processes with semi-Markov switching to elucidate the impact of common characteristics of the investors on the aggregate quantities like the market prices.Optimal stopping problems have applications in the areas of seismology, machine monitoring, finance, insurance, health surveillance among others. Improved stochastic control techniques may inform the public, financial planners and legislators about the risk of ruin in retirement. This work will benefit individuals' decision making on important financial matters they face during their lives: How much to invest in mutual funds; how much insurance one should buy; whether it is a good time to borrow to invest in the stock market and how much one should borrow; when it is best to declare bankruptcy; when one should retire, etc. The development of new pricing principles provides better pricing mechanisms for derivative products in the markets, which will benefit the financial institutions which are at the center of the economy. The results in the final section of this part will help the management make better decisions for the welfare of their companies which will benefit the nation since available resources will be used more efficiently. The final objective of the project will obtain insights into the market dynamics by understanding the price formation from typical behavioral qualities of investors. This is important in creating good financial models that benefit economic forecast, investment and policy decisions.