Computational schemes for stochastic singular control problems, with applications in portfolio optimization

Singular stochastic control problems are those problems in which control effort can effect instantaneous displacement in state. Such problems find diverse applications in operations management, economics and finance. Yet, in all but the simplest of cases, singular stochastic control problems cannot be solved analytically. This thesis provides computational schemes to solve control problems that have singular controls in them. It is divided into two parts.;Part I proposes a method for numerically solving, a class of singular stochastic control problems, that are primarily characterized by having; only singular controls. We combine finite element methods that numerically solve partial differential equations with a policy update procedure based on the principle of smooth pasting to iteratively solve Hamilton-Jacobi-Bellman equations associated with the stochastic control problem. We illustrate the method on two examples of singular stochastic control problems, one drawn from queueing systems and the other from economics.;The multi-dimensional portfolio optimization problem with transaction costs is the focus in part II. It belongs to the class of stochastic control problems that have both singular as well as continuous controls. Specifically, it considers the problem of optimally allocating wealth among multiple stocks and a bank account, in order to maximize the infinite horizon discounted utility of consumption. The transfer of wealth from one asset to another involves proportional transaction costs. The model allows for correlation between the price processes. Adapting the finite element method and using an iterative procedure that converts the free-boundary problem into a sequence of fixed boundary problems; we provide an efficient computational scheme for solving this free boundary problem. We present results that describe dependence of optimal policy to change in model parameters. Finally we suggest and quantify certain heuristic approximations for the optimal policy.