A Convex Stochastic Optimization Problem Arising from Portfolio Selection

13 Pages Posted: 19 Dec 2007

See all articles by Hanqing Jin

Hanqing Jin

Chinese University of Hong Kong

Zuo Quan Xu

affiliation not provided to SSRN

Xun Yu Zhou

University of Toronto

Abstract

A continuous-time financial portfolio selection model with expected utility maximization typically boils down to solving a (static) convex stochastic optimization problem in terms of the terminal wealth, with a budget constraint. In literature the latter is solved by assuming a priori that the problem is well-posed (i.e., the supreme value is finite) and a Lagrange multiplier exists (and as a consequence the optimal solution is attainable). In this paper it is first shown that, via various counter-examples, neither of these two assumptions needs to hold, and an optimal solution does not necessarily exist. These anomalies in turn have important interpretations in and impacts on the portfolio selection modeling and solutions. Relations among the non-existence of the Lagrange multiplier, the ill-posedness of the problem, and the non-attainability of an optimal solution are then investigated. Finally, explicit and easily verifiable conditions are derived which lead to finding the unique optimal solution.

Suggested Citation

Jin, Hanqing and Quan Xu, Zuo and Yu Zhou, Xun, A Convex Stochastic Optimization Problem Arising from Portfolio Selection. Mathematical Finance, Vol. 18, Issue 1, pp. 171-183, January 2008, Available at SSRN: https://ssrn.com/abstract=1073261 or http://dx.doi.org/10.1111/j.1467-9965.2007.00327.x

Hanqing Jin (Contact Author)

Chinese University of Hong Kong ( email )

Hong Kong
Hong Kong

Zuo Quan Xu

affiliation not provided to SSRN

Xun Yu Zhou

University of Toronto

Toronto, Ontario M5S 3G8
Canada

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