Optimization of Expected Utility of Consumption and Terminal wealth in a portfolio in continuous time.

Abstract

In this dissertation, addressing an investment-consumption problem with the goal of maximizing expected utility of consumption and terminal wealth in a portfolio in continuous time is considered. The action is two-fold in which one has to periodically make decisions on asset-allocation and consumption. The investor considered is retired, therefore accumulates wealth through investment while also consuming it along the way but not exceeding the wealth at hand until some terminal point in time. The objective is to maximize the expected utility derived from consumption and terminal wealth. Using the Hamilton-Jacobi-Bellman (HJB) equations, the study analytically investigates a special case of stochastic optimal control problem where the state equations are linear in both the state and control, and the cost functional is non-linear done in the setting of a Black Scholes market model. The optimal portfolio processes and expected wealth are established and with these the Optimal Value Function which is the objective function is obtained. In conclusion, the optimal behavior is to consume modestly and invest more when one is younger, then to gradually increase the consumption as one ages.