## The Annals of Applied Probability

### 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 $\mathscr{R}^d$. 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

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

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aoap/1177005576

Digital Object Identifier
doi:10.1214/aoap/1177005576

Mathematical Reviews number (MathSciNet)
MR1189418

Zentralblatt MATH identifier
0770.90002

JSTOR