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Convex duality in constrained mean-variance portfolio optimization

Part of: Miscellaneous topics in calculus of variations and optimal control Stochastic systems and control

Published online by Cambridge University Press:  01 July 2016

Chantal Labbé  and
Andrew J. Heunis
Chantal Labbé*
Affiliation:
HEC Montréal
Andrew J. Heunis*
Affiliation:
University of Waterloo
*
∗ Postal address: HEC Montréal, Montréal, QC H3T 2A7, Canada. Email address: chantal.labbe@hec.ca
∗∗ Postal address: Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, ON N2L 3G1, Canada. Email address: heunis@kingcong.uwaterloo.ca
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Abstract

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We apply conjugate duality to establish the existence of optimal portfolios in an asset-allocation problem, with the goal of minimizing the variance of the final wealth which results from trading over a fixed, finite horizon in a continuous-time, complete market, subject to the constraints that the expected final wealth equal a specified target value and the portfolio of the investor (defined by the dollar amount invested in each stock) take values in a given closed, convex set. The asset prices are modelled by Itô processes, for which the market parameters are random processes adapted to the information filtration available to the investor. We synthesize a dual optimization problem and establish a set of optimality relations, similar to the Euler-Lagrange and transversality relations of calculus of variations, giving necessary and sufficient conditions for the given optimization problem and its dual to each have a solution, with zero duality gap. We then solve these relations, to establish the existence of an optimal portfolio.

Keywords

Convex analysisduality synthesisvariational analysis

MSC classification

Primary: 93E20: Optimal stochastic control 49N15: Duality theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 39 , Issue 1 , March 2007 , pp. 77 - 104
Copyright
Copyright © Applied Probability Trust 2007 

Footnotes

∗∗∗

Supported by the National Sciences and Engineering Ressearch Council of Canada.

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