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

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