Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-27T10:28:37.933Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

References

Aubin, J.-P. (1978). Applied Functional Analysis. John Wiley, New York.Google Scholar
Bismut, J.-M. (1973). Conjugate convex functions in optimal stochastic control. J. Math. Anal. Appl. 44, 384–404.Google Scholar
Chow, Y. S. and Teicher, H. (1988). Probability Theory: Independence, Interchangeability, Martingales, 2nd edn. Springer, New York.Google Scholar
Cuoco, D. and Liu, H. (2000). A martingale characterization of consumption choices and hedging costs with margin requirements. Math. Finance 10, 355–385.Google Scholar
Cvitanić, J. and Karatzas, I. (1992). Convex duality in constrained portfolio optimization. Ann. Appl. Prob. 2, 767–818.Google Scholar
Dubovitskii, A. Ya. and Miĺyutin, A. A. (1968). Necessary conditions for a weak extremum in problems of optimal control with mixed inequality constraints. Zhur. Vychislitel. Mat. Mat. Fys. 8, 725–779.Google Scholar
Ekeland, I. and Témam, R. (1999). Convex Analysis and Variational Problems (Classics Appl. Math. 88). Society for Industrial and Applied Mathematics, Philadelphia, PA.Google Scholar
Karatzas, I. and Shreve, S. E. (1988). Brownian Motion and Stochastic Calculus. Springer, New York.CrossRefGoogle Scholar
Karatzas, I. and Shreve, S. E. (1998). Methods of Mathematical Finance. Springer, New York.Google Scholar
Karatzas, I., Lehoczky, J. P., Shreve, S. E. and Xu, G. L. (1991). Martingale and duality methods for utility maximization in an incomplete market. SIAM J. Control Optimization 29, 702–730.CrossRefGoogle Scholar
Labbé, C. (2004). Contributions to the theory of constrained portfolio optimization. , Department of Statistics and Actuarial Sciences, University of Waterloo.Google Scholar
Li, X., Zhou, X. Y. and Lim, A. E. B. (2002). Dynamic mean-variance portfolio selection with no-shorting constraints. SIAM J. Control Optimization 40, 1540–1555.CrossRefGoogle Scholar
Lim, A. E. B. and Zhou, X. Y. (2002). Mean-variance portfolio selection with random parameters in a complete market. Math. Operat. Res. 27, 101–120.Google Scholar
Rogers, L. C. G. (2003). Duality in constrained optimal investment and consumption problems: a synthesis. In Paris–Princeton Lectures on Mathematical Finance, 2002 (Lecture Notes Math. 1814), Springer, Berlin, pp. 95–131.Google Scholar
Vinter, R. B. (2000). Optimal Control. Birkhäuser, Boston, MA.Google Scholar
Yong, J. and Zhou, X. Y. (1999). Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer, New York.Google Scholar