Multivariate utility maximization with proportional transaction costs

TL;DR

This paper establishes an optimal investment framework in a currency exchange setting with proportional transaction costs, utilizing multivariate utility functions and proving the existence of optimal portfolios under certain conditions.

Contribution

It introduces a new optimal investment theorem accommodating multivariate utilities and discontinuous transaction costs, extending prior models to more realistic currency exchange scenarios.

Findings

Existence of optimal portfolio processes under asymptotic satiability.

Reformulation of the optimization problem in terms of terminal liquidation utility.

Conditions ensuring asymptotic satiability include utility elasticity and dual function growth.

Abstract

We present an optimal investment theorem for a currency exchange model with random and possibly discontinuous proportional transaction costs. The investor's preferences are represented by a multivariate utility function, allowing for simultaneous consumption of any prescribed selection of the currencies at a given terminal date. We prove the existence of an optimal portfolio process under the assumption of asymptotic satiability of the value function. Sufficient conditions for asymptotic satiability of the value function include reasonable asymptotic elasticity of the utility function, or a growth condition on its dual function. We show that the portfolio optimization problem can be reformulated in terms of maximization of a terminal liquidation utility function, and that both problems have a common optimizer.