Title: On utility maximization without passing by the dual problem,
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.