Ergodic robust maximization of asymptotic growth

TL;DR

This paper develops a robust framework for maximizing long-term investment growth under model uncertainty, identifying optimal strategies and growth rates using large deviations theory, with applications to equity market capitalizations.

Contribution

It introduces a novel approach to robust growth maximization by linking it to Donsker-Varadhan rate functions and explicitly characterizing optimal trading strategies.

Findings

Robust growth rate equals the Donsker-Varadhan rate function.

Existence and explicit form of the optimal trading strategy.

Application to ranked market capitalizations with explicit strategies.

Abstract

We consider the problem of robustly maximizing the growth rate of investor wealth in the presence of model uncertainty. Possible models are all those under which the assets' region $E$ and instantaneous covariation $c$ are known, and where additionally the assets are stable in that their occupancy time measures converge to a law with density $p$. This latter assumption is motivated by the observed stability of ranked relative market capitalizations for equity markets. We seek to identify the robust optimal growth rate, as well as a trading strategy which achieves this rate in all models. Under minimal assumptions upon $(E,c,p)$, we identify the robust growth rate with the Donsker-Varadhan rate function from occupancy time Large Deviations theory. We also prove existence of, and explicitly identify, the optimal trading strategy. We then apply our results in the case of drift uncertainty for ranked relative market capitalizations. Assuming regularity under symmetrization for the covariance and limiting density of the ranked capitalizations, we explicitly identify the robust optimal trading strategy in this setting.