Explicit Solution of a General Consumption/Investment Problem

Abstract

This paper solves a general consumption and investment decision problem in closed form. An investor seeks to maximize total expected discounted utility of consumption. There are N distinct risky investments, modeled by dependent geometric Brownian motion processes, and one risk-less (deterministic) investment. The analysis allows for a general utility function and general rates of return. The model and analysis take into consideration the inherent non-negativity of consumption and consider bankruptcy, so this paper generalizes many of the results of Lehoczky, Sethi, and Shreve. The value function is determined explicitly, as are the optimal consumption and investment policies. The analysis is extended to consider more general risky investments. Under certain conditions, the value functions derived for geometric Brownian motion are shown to provide upper and lower bounds on the value functions in the more general context.

Keywords: Mutual fund theorem,investment-consumption problem, consumption/portfolio problem, dynamic programming, stochastic control, bankruptcy,Brownian motion

JEL Classification: C61, E21,G11, G12