A convex stochastic optimization problem arising from portfolio selection

Hanqing Jin, Zuo Quan Xu, Xun Yu Zhou

Research output: Journal article publicationJournal articleAcademic researchpeer-review

57 Citations (Scopus)


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 supremum 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.

Original languageEnglish
Pages (from-to)171-183
Number of pages13
JournalMathematical Finance
Issue number1
Publication statusPublished - Jan 2008
Externally publishedYes


  • Attainability
  • Convex stochastic optimization
  • Lagrange multiplier
  • Portfolio selection
  • Well-posedness

ASJC Scopus subject areas

  • Accounting
  • Social Sciences (miscellaneous)
  • Finance
  • Economics and Econometrics
  • Applied Mathematics


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