On utility maximization without passing by the dual problem

TL;DR

This paper establishes the existence of optimal investment strategies in continuous-time financial markets with unbounded endowments and non-smooth utilities without relying on dual problems, using Orlicz space theory.

Contribution

It provides a novel proof of utility maximization existence that avoids duality, accommodating non-smooth utilities, non-hedgeable risks, and terminal wealth constraints.

Findings

Existence of optimal investment strategies proven without dual problem.

Applicable to non-smooth and strictly concave utilities.

Handles unbounded endowments and constraints in various market models.

Abstract

We treat utility maximization from terminal wealth for an agent with utility function $U:\mathbb{R}\to\mathbb{R}$ who dynamically invests in a continuous-time financial market and receives a possibly unbounded random endowment. We prove the existence of an optimal investment without introducing the associated dual problem. We rely on a recent result of Orlicz space theory, due to Delbaen and Owari which leads to a simple and transparent proof. Our results apply to non-smooth utilities and even strict concavity can be relaxed. We can handle certain random endowments with non-hedgeable risks, complementing earlier papers. Constraints on the terminal wealth can also be incorporated. As examples, we treat frictionless markets with finitely many assets and large financial markets.