A Convex Stochastic Optimization Problem Arising from Portfolio Selection

TL;DR

This paper examines the convex stochastic optimization problem in portfolio selection, revealing that common assumptions of well-posedness and solution attainability may fail, and provides conditions for finding the unique optimal portfolio.

Contribution

It challenges standard assumptions in portfolio optimization, showing that optimal solutions may not exist and offering explicit conditions for their existence.

Findings

Counter-examples show assumptions can fail

Relations among non-existence, ill-posedness, and non-attainability

Explicit conditions for unique optimal solutions

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.