Utility maximization problem under transaction costs: optimal dual processes and stability

TL;DR

This paper investigates the stability and properties of optimal dual processes in utility maximization problems under transaction costs, establishing convergence results and the existence of shadow prices in continuous markets.

Contribution

It provides a detailed analysis of the stability of primal and dual solutions and constructs optimal dual processes in limiting markets, introducing new methods for handling perturbations.

Findings

Stability of primal and dual value functions under perturbations.

Convergence of primal and dual optimizers when utility or probability measures change.

Construction of shadow prices via limiting optimal dual processes in continuous markets.

Abstract

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.