A unified framework for utility maximization problems: An Orlicz space approach

TL;DR

This paper develops a unified mathematical framework using Orlicz spaces for utility maximization in incomplete financial markets with possibly unbounded price processes, allowing for broader utility functions and strategy classes.

Contribution

It introduces a novel Orlicz space approach to unify utility maximization problems for different utility domains and proves the existence of solutions via duality methods.

Findings

Existence of primal and dual solutions in the generalized setting

Inclusion of utility functions finite on the entire real line

Recognition of potential singular components in pricing functionals

Abstract

We consider a stochastic financial incomplete market where the price processes are described by a vector-valued semimartingale that is possibly nonlocally bounded. We face the classical problem of utility maximization from terminal wealth, with utility functions that are finite-valued over $(a,\infty)$, $a\in\lbrack-\infty,\infty)$, and satisfy weak regularity assumptions. We adopt a class of trading strategies that allows for stochastic integrals that are not necessarily bounded from below. The embedding of the utility maximization problem in Orlicz spaces permits us to formulate the problem in a unified way for both the cases $a\in\mathbb{R}$ and $a=-\infty$. By duality methods, we prove the existence of solutions to the primal and dual problems and show that a singular component in the pricing functionals may also occur with utility functions finite on the entire real line.