Robust maximization of asymptotic growth

TL;DR

This paper develops a robust investment strategy that maximizes long-term growth in markets with unknown expected returns, using a generalized eigenfunction approach tied to the process's covariance structure.

Contribution

It introduces a novel method for robust growth optimization based on a generalized principal eigenfunction for elliptic operators, extending previous arbitrage concepts.

Findings

Identifies the optimal investment strategy using a generalized eigenfunction.

Connects the strategy to limits of optimal arbitrages as time horizon extends.

Provides a framework for robust growth maximization under model uncertainty.

Abstract

This paper addresses the question of how to invest in a robust growth-optimal way in a market where the instantaneous expected return of the underlying process is unknown. The optimal investment strategy is identified using a generalized version of the principal eigenfunction for an elliptic second-order differential operator, which depends on the covariance structure of the underlying process used for investing. The robust growth-optimal strategy can also be seen as a limit, as the terminal date goes to infinity, of optimal arbitrages in the terminology of Fernholz and Karatzas [Ann. Appl. Probab. 20 (2010) 1179-1204].