The Annals of Applied Probability

Convex Duality in Constrained Portfolio Optimization

Jaksa Cvitanic and Ioannis Karatzas

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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 $\mathscr{R}^d$. The setting is that of a continuous-time, Ito 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.

Article information

Ann. Appl. Probab., Volume 2, Number 4 (1992), 767-818.

First available in Project Euclid: 19 April 2007

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Zentralblatt MATH identifier


Primary: 93E20: Optimal stochastic control
Secondary: 90A09 60H30: Applications of stochastic analysis (to PDE, etc.) 60G44: Martingales with continuous parameter 90A16 49N15: Duality theory

Constrained optimization convex analysis duality stochastic contro portofolio and consumption processes martingale representations


Cvitanic, Jaksa; Karatzas, Ioannis. Convex Duality in Constrained Portfolio Optimization. Ann. Appl. Probab. 2 (1992), no. 4, 767--818. doi:10.1214/aoap/1177005576.

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