Some optimal control problems in mathematical finance

We study several optimal control problems arising in Mathematical Finance. The first problem is related to models of Term Structure of Interest Rates, which is central to economic and financial theory and provides the basis for pricing interest rate derivatives. Within the framework of optimal control for infinite dimension systems, we develop new approaches to estimate the risk neutral drift term for both the one and the multi-factor models. Second, in pricing derivatives, a key and often difficult issue is how to deal with stochastic volatility. We show that our new techniques can also be employed to address this problem. Third, we investigate the problem of building envelopes for the values of convertible bonds under stochastic interest rates. Finally, using our techniques, we examine the bond pricing problem with the presence of default risk, a topic that has received much attention in recent years. The related optimal control problems are analyzed mathematically and conditions for optimality are obtained via a variational approach. Also, numerical algorithms are proposed and tested, and computational results are obtained.