Topics in Stochastic Control with Applications to Finance.

Abstract

This thesis is devoted to PDE characterization for stochastic control problems when the classical methodology of dynamic programming does not work. Under the framework of viscosity solutions, a dynamic programming principle (DPP) serves as the tool to associate a (nonlinear) PDE to a stochastic control problem. Unfortunately, a DPP is in general difficult to prove, and may fail to be true in some cases. In this thesis, we investigate three different scenarios where classical dynamic programming does not work. The first one is quantile hedging in the presence of arbitrage, the second one is robust growth-optimal trading, and the third one is a stochastic differential game of control and stopping. In each of the cases, we develop new methods to circumvent the lack of a classical DPP.