The first part of the project will consider optimal stopping problems in which the well-known results of the optimal stopping theory do not apply. An example is the American option pricing problem when there are weak arbitrage opportunities or a stock price bubble. The problems in this part are related to determining necessary and sufficient conditions for the uniqueness of solutions of the so-called Cauchy problems. Quantile hedging (hedging with high probability of success) problems, solutions of which are characterized by non-linear partial differential equations, will also be investigated in the above context. In the second part, I will develop a new theory for optimal stopping problems when the statistical expectation operator is replaced by alternative ways of measuring future rewards, for example by the so-called risk measures. These problems are related to stochastic differential games of control and stopping. Saddle points of such games will be determined. In some special cases, these are related free boundaries with quasi-linear (integro) partial differential equations. Regularity of the value function and the free boundary curve will be investigated. The relationship between the risk aversion and the shape of the free boundary will also be analyzed. In the third part, pathwise comparison and convex duality methods will be used to solve optimization problems with objectives of maximizing the probability of reaching certain goals. Utility maximization problems with risk constraints that will be considered in this part are quite relevant given the current economic environment in which large investors face regulatory risk constraints.
The project will lead to several methodological/theoretical developments in the theory of optimal stopping and stochastic control. As a by-product, these developments will help us understand the existence, uniqueness, and regularity questions in linear/non-linear partial differential equations. Our results will resolve important pricing and hedging questions in Mathematical Finance. This project complements the recent developments in the theory of risk measures by addressing decision making problems using these measures as optimization criteria, which is an important step in managing financial risk.
