Explicit Solution of a General Consumption/Investment Problem

Mathematics of Operations Research, Vol. 11, No. 2, pp. 261-294, May 1986

34 Pages Posted: 22 Jan 2008 Last revised: 7 Nov 2015

See all articles by Ioannis Karatzas

Ioannis Karatzas

Columbia University - Department of Statistics

John P. Lehoczky

Carnegie Mellon University

Suresh Sethi

University of Texas at Dallas - Naveen Jindal School of Management

Steven E. Shreve

Carnegie Mellon University - Department of Mathematical Sciences

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

Suggested Citation

Karatzas, Ioannis and Lehoczky, John and Sethi, Suresh and Shreve, Steven E., Explicit Solution of a General Consumption/Investment Problem. Mathematics of Operations Research, Vol. 11, No. 2, pp. 261-294, May 1986, Available at SSRN: https://ssrn.com/abstract=1086184

Ioannis Karatzas

Columbia University - Department of Statistics ( email )

Mail Code 4403
2990 Broadway, Room 618
New York, NY 10027
United States
212-854-3177 (Phone)
212-663-2454 (Fax)

John Lehoczky

Carnegie Mellon University ( email )

Pittsburgh, PA 15213-3890
United States

Suresh Sethi (Contact Author)

University of Texas at Dallas - Naveen Jindal School of Management ( email )

800 W. Campbell Road, SM30
Richardson, TX 75080-3021
United States

Steven E. Shreve

Carnegie Mellon University - Department of Mathematical Sciences ( email )

Pittsburgh, PA 15213-3890
United States

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