Duality Theory for Robust Utility Maximisation

TL;DR

This paper develops a duality framework for robust utility maximisation in continuous time, extending classical results to account for model uncertainty in trading outcomes and pricing measures.

Contribution

It introduces a duality theory for robust utility maximisation under uncertainty, including existence of optimal strategies and applicability to various utility functions.

Findings

Duality holds under bipolar relations between trading outcomes and pricing measures.

Results include cases of logarithmic and power utility functions.

Applicable to models with drift and volatility uncertainty.

Abstract

In this paper we present a duality theory for the robust utility maximisation problem in continuous time for utility functions defined on the positive real axis. Our results are inspired by -- and can be seen as the robust analogues of -- the seminal work of Kramkov & Schachermayer [18]. Namely, we show that if the set of attainable trading outcomes and the set of pricing measures satisfy a bipolar relation, then the utility maximisation problem is in duality with a conjugate problem. We further discuss the existence of optimal trading strategies. In particular, our general results include the case of logarithmic and power utility, and they apply to drift and volatility uncertainty.