Convex Duality in Constrained Portfolio Optimization

Abstract

We study the stochastic control problem of maximizing expected utility from terminal wealth and/or consumption, when the portfolio is constrained to take values in a given closed, convex subset of R^d. The setting is that of a continuous-time, Itô process model for the underlying asset prices. General existence results are established for optimal portfolio/consumption strategies, by suitably embedding the constrained problem in an appropriate family of unconstrained ones, and finding a member of this family for which the corresponding optimal policy obeys the constraints. Equivalent conditions for optimality are obtained, and explicit solutions leading to feedback formulae are derived for special utility functions and for deterministic coefficients. Results on incomplete markets, on short-selling constraints and on different interest rates for borrowing and lending are covered as special cases. The mathematical tools are those of continuous-time martingales, convex analysis and duality theory.

Additional Information

© 1992 Institute of Mathematical Statistics. Received July 1991; revised November 1991. The results of this paper have been drawn from the first author's doctoral dissertation at Columbia. A preliminary version was presented as an invited lecture at the 20th Conference on Stochastic Processes and their Applications, Nahariya, Israel, June 1991. Research supported in part by NSF Grant DMS-90-22188. We wish to thank Jérôme Detemple, Ralf Korn and the two anonymous referees, for their careful reading of the paper and their helpful suggestions.

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