🤖 AI Summary
This paper studies the robust asymptotic growth problem under stochastic factor dynamics: the asset price process (X) is driven by a non-Markovian—and even non-semimartingale—stochastic factor (Y), with drift uncertainty in (X) and local model uncertainty in (Y), while fixing both the instantaneous covariance structure of (X) and the joint invariant density of ((X,Y)). Methodologically, it integrates PDE analysis, variational methods, and generalized Dirichlet form theory, relaxing classical Markovian and semimartingale assumptions. It provides, for the first time in a non-Markovian setting with non-semimartingale factors, a complete characterization of the robust optimal growth rate and constructs an explicit, factor-path-independent functional-generated trading strategy—fully resolving the central question posed by Fernholz (2002). The results substantially extend the Kardaras–Robertson framework, delivering explicit growth-rate representations and robust strategy constructions under weak regularity conditions.
📝 Abstract
We consider an asymptotic robust growth problem under model uncertainty and in the presence of (non-Markovian) stochastic covariance. We fix two inputs representing the instantaneous covariance for the asset 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. Mathematically, we use a combination of partial differential equation (PDE), calculus of variations and generalized Dirichlet form techniques.
