Necessary Optimality Conditions for Two Stochastic Control Problems

This thesis consists of two papers concerning necessary conditions in stochas-tic control problems. In the first paper, we study the problem of controlling a linear stochastic differential equation (SDE) where the coefficients are random and not necessarily bounded. We consider relaxed control processes, i.e. the control is defined as a process taking values in the space of probability measures on the control set. The main motivation is a bond portfolio optimization prob-lem. The relaxed control processes are then interpreted as the portfolio weights corresponding to different maturity times of the bonds. We establish existence of an optimal control and necessary conditons for optimality in the form of a maximum principle, extended to include the family of relaxed controls. In the second paper we consider the so-called singular control problem where the control consists of two components, one absolutely continuous and one sin-gular. The absolutely continuous part of the control is allowed to enter both the drift and diffusion coefficient. The absolutely continuous part is relaxed i