  This paper discusses the num\'eraire-based utility maximization problem in
markets with proportional transaction costs. In particular, the investor is
required to liquidate all her position in stock at the terminal time. We first
observe the stability of the primal and dual value functions as well as the
convergence of the primal and dual optimizers when perturbations occur on the
utility function and on the physical probability. We then study the properties
of the optimal dual process (ODP), that is, a process from the dual domain that
induces the optimality of the dual problem. When the market is driven by a
continuous process, we construct the ODP for the problem in the limiting market
by a sequence of ODPs corresponding to the problems with small misspecificated
parameters. Moreover, we prove that this limiting ODP defines a shadow price.
