On utility maximization under convex portfolio constraints

TL;DR

This paper investigates a broad utility-maximization framework in financial models with convex portfolio constraints, proving the existence of optimal strategies without smoothness assumptions and simplifying the dual problem formulation.

Contribution

It extends classical utility maximization to include non-origin containing convex constraints and demonstrates the existence of optimal strategies under minimal utility smoothness.

Findings

Optimal trading strategies exist under general convex constraints.

Dual problem can be formulated over countably-additive measures without enlargement.

Framework includes models with illiquid assets and random endowments.

Abstract

We consider a utility-maximization problem in a general semimartingale financial model, subject to constraints on the number of shares held in each risky asset. These constraints are modeled by predictable convex-set-valued processes whose values do not necessarily contain the origin; that is, it may be inadmissible for an investor to hold no risky investment at all. Such a setup subsumes the classical constrained utility-maximization problem, as well as the problem where illiquid assets or a random endowment are present. Our main result establishes the existence of optimal trading strategies in such models under no smoothness requirements on the utility function. The result also shows that, up to attainment, the dual optimization problem can be posed over a set of countably-additive probability measures, thus eschewing the need for the usual finitely-additive enlargement.