  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.

Mathematical Finance;Probability