Convex Duality in Constrained Portfolio Optimization

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. We study the stochastic control problem of maximizing expected utility from terminal wealth and/or consumption, when the portfolio is con-strained to take values in a given closed, convex subset of Md. The setting is that of a continuous-time, Ito process model for the underlying asset prices. General existence results are established for optimal portfolio/con-sumption 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-sell-ing constraints and on different interest rates for borrowing and lending are covered as special cases. The mathematical tools are those of continu-ous-time martingales, convex analysis and duality theory.