Ergodic robust maximization of asymptotic growth with stochastic factor processes
David Itkin, Benedikt Koch, Martin Larsson, Josef Teichmann
TL;DR
The paper tackles robust asymptotic growth optimization under model uncertainty in markets with ergodic dynamics and a stochastic factor Y that influences instantaneous covariance. It extends Kardaras & Robertson by allowing a factor-driven covariance c_X(X,Y), and shows the robust optimal trading strategy is functionally generated, depending only on the asset prices X, with a PDE/variational characterization via an optimal φ̂. A worst-case measure is constructed within a broad class of models using generalized Dirichlet forms, establishing that the robust growth rate λ is given by λ=1/2 ∫_E ∇φ̂^⊤ A ∇φ̂ and that the optimal strategy θ̂=∇φ̂(X) is invariant to Y, even when the joint covariation is fully specified. The framework yields K-modifications to guarantee ergodicity, but the main results are robust to the choice of K and extend to various concrete examples, including gradient, one-dimensional, tractable model classes, exogenous factors, and Beta-density settings. Collectively, the work provides a comprehensive, theory-grounded answer to Fernholz’s stochastic portfolio theory question on growth-optimal, functionally generated strategies under ergodic, factor-influenced model uncertainty with broad applicability.
Abstract
We consider a robust asymptotic growth problem under model uncertainty in the presence of stochastic factors. We fix two inputs representing the instantaneous covariance for the asset price process $X$, which depends on an additional stochastic factor process $Y$, as well as the invariant density of $X$ together with $Y$. The stochastic factor process $Y$ has continuous trajectories but is not even required to be a semimartingale. Our setup allows for drift uncertainty in $X$ and model uncertainty for the local dynamics of $Y$. This work builds upon a recent paper of Kardaras & Robertson, where the authors consider an analogous problem, however, without the additional stochastic factor process. Under suitable, quite weak assumptions we are able to characterize the robust optimal trading strategy and the robust optimal growth rate. The optimal strategy is shown to be functionally generated and, remarkably, does not depend on the factor process $Y$. Our result provides a comprehensive answer to a question proposed by Fernholz in 2002. We also show that the optimal strategy remains optimal even in the more restricted case where $Y$ is a semimartingale and the joint covariation structure of $X$ and $Y$ is prescribed as a function of $X$ and $Y$. Our results are obtained using a combination of techniques from partial differential equations, calculus of variations, and generalized Dirichlet forms.