On Merton's Portfolio Problem : A Stochastic Optimal Control Problem

Abstract: The purpose of this thesis is to examine and solve a classic financial optimization problem known as Merton’s Portfolio Problem. The problem is driven by a stochastic process and can thereby be classified as a stochastic optimal control problem. The thesis provides the basic theory of how the dynamic programming principle allows dividing the control problem into subproblems that can be linked to a PDE. The solution to the PDE can then be shown to rely on the optimal control sought after in the stochastic control problem. Merton’s Portfolio Problem is presented and solved. The explicit solution and the financial implications are then examined, both analytically and by looking at some simulated trajectories.