A note on utility maximization with transaction costs and random endoment: num\'eraire-based model and convex duality

TL;DR

This paper extends the utility maximization framework under proportional transaction costs and random endowment using convex duality in a numéraire-based model, broadening previous results to include finitely additive measures.

Contribution

It develops a convex duality approach for utility maximization with transaction costs, enlarging the dual domain to finitely additive measures, extending prior work to more general settings.

Findings

Established convex duality results under transaction costs.

Extended dual domain to finitely additive measures.

Connected to and extended prior foundational work.

Abstract

In this note, we study the utility maximization problem on the terminal wealth under proportional transaction costs and bounded random endowment. In particular, we restrict ourselves to the num\'eraire-based model and work with utility functions only supporting R+. Under the assumption of existence of consistent price systems and natural regularity conditions, standard convex duality results are established. Precisely, we first enlarge the dual domain from the collection of martingale densities associated with consistent price systems to a set of finitely additive measures; then the dual formulation of the utility maximization problem can be regarded as an extension of the paper of Cvitani\'c-Schachermayer-Wang (2001) to the context under proportional transaction costs.