Utility Maximization Problem with Transaction Costs: Optimal Dual Processes and Stability

In this paper, we investigate the stability problem of the numeraire-based utility maximization problem in markets with transaction costs, where the stock price is not necessarily a semimartingale. Precisely, the static stability of primal and dual value functions as well as the convergence of primal and dual optimizers are presented when perturbations occur in the utility function and in the physical probability. Furthermore, this study focuses on the optimal dual process (ODP), which induces the dual optimizer and attains optimality for a dynamical dual problem. Properties of ODPs are discussed which are complement of the duality theory for this utility maximization problem. When the parameters of the market and the investor are slightly perturbed, both the dual optimizer and the associated optimal dual process are stable. Thus, a shadow price process is constructed based on the sequence of ODPs corresponding to problems with small misspecified parameters.