Duality Theory for Portfolio Optimisation under Transaction Costs

TL;DR

This paper develops a duality theory for portfolio optimization with proportional transaction costs, introducing a shadow price process defined via a 'sandwiched' supermartingale, applicable to general cadlag price processes.

Contribution

It establishes the existence of a dual optimizer and a generalized shadow price process for portfolio optimization under transaction costs in a broad setting.

Findings

Existence of dual optimizer proven.

Shadow price process characterized as a 'sandwiched' supermartingale.

Applicable to general cadlag price processes.

Abstract

For portfolio optimisation under proportional transaction costs, we provide a duality theory for general cadlag price processes. In this setting, we prove the existence of a dual optimiser as well as a shadow price process in a generalised sense. This shadow price is defined via a "sandwiched" process consisting of a predictable and an optional strong supermartingale and pertains to all strategies which remain solvent under transaction costs. We provide examples showing that in the present general setting the shadow price process has to be of this generalised form.