On utility maximization without passing by the dual problem

Abstract

We treat utility maximization from terminal wealth for an agent with utility function 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.

Keywords:

• Utility function
• optimal investment
• unbounded random endowments
• transaction costs
• large markets

Acknowledgements

The author thanks Freddy Delbaen and Keita Owari for discussions about Section 2 and an anonymous referee for very useful comments that led to substantial improvements. Special thanks go to Ngoc Huy Chau for discussions which helped discovering and removing an error.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The support received from the ‘Lendület’ grant LP 2015-6 of the Hungarian Academy of Sciences and from the NKFIH (National Research, Development and Innovation Office, Hungary) grant KH 126505 is gratefully acknowledged.