  This paper studies the utility maximization on the terminal wealth with
random endowments and proportional transaction costs. To deal with unbounded
random payoffs from some illiquid claims, we propose to work with the
acceptable portfolios defined via the consistent price system (CPS) such that
the liquidation value processes stay above some stochastic thresholds. In the
market consisting of one riskless bond and one risky asset, we obtain a type of
super-hedging result. Based on this characterization of the primal space, the
existence and uniqueness of the optimal solution for the utility maximization
problem are established using the duality approach. As an important application
of the duality theorem, we provide some sufficient conditions for the existence
of a shadow price process with random endowments in a generalized form as well
as in the usual sense using acceptable portfolios.
