A Convex Stochastic Optimization Problem Arising from Portfolio Selection

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 {\it 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, 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.

15 pages

Cited by in corpus (6)

• A Note on the Quantile Formulation
• Optimal stopping under probability distortion
• Dynamic Mean-LPM and Mean-CVaR Portfolio Optimization in Continuous-time
• Optimal management of DC pension fund under relative performance ratio and VaR constraint
• Behavioral Portfolio Selection in Continuous Time
• Investment under Duality Risk Measure