Stability of utility-maximization in incomplete markets

TL;DR

This paper examines the stability of utility-maximization strategies in incomplete markets, analyzing how small changes in market parameters affect optimal portfolio choices and establishing conditions for continuity.

Contribution

It identifies topologies ensuring the continuity of utility-maximization with respect to market parameters and provides a counterexample demonstrating the limits of this stability.

Findings

Utility-maximization is continuous under specific topologies.

Counterexample shows the limits of stability in certain settings.

New structural results for solutions with continuous semimartingale prices.

Abstract

The effectiveness of utility-maximization techniques for portfolio management relies on our ability to estimate correctly the parameters of the dynamics of the underlying financial assets. In the setting of complete or incomplete financial markets, we investigate whether small perturbations of the market coefficient processes lead to small changes in the agent's optimal behavior derived from the solution of the related utility-maximization problems. Specifically, we identify the topologies on the parameter process space and the solution space under which utility-maximization is a continuous operation, and we provide a counterexample showing that our results are best possible, in a certain sense. A novel result about the structure of the solution of the utility-maximization problem where prices are modeled by continuous semimartingales is established as an offshoot of the proof of our central theorem.