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, modelled by dependent geometric Brownian motion processes, and one riskless (deterministic) investment. The analysis allows for a general utility function and general rates of return. The model and analysis take into consideration the inherent nonnegativity of consumption and consider bankruptcy, so this paper generalizes many of the results of Lehoczky, Sethi, and Shreve (Lehoczky, J., S. Sethi, S. Shreva. 1983. Optimal consumption and investment policies allowing consumption constraints and bankruptcy. Math. Oper. Res. 8 613–636.). 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.

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