Signature varieties and rough paths

Abstract: We study signature tensors of paths from an algebraic geometric viewpoint. The signatures of a given class of paths parametrize a variety inside the space of tensors. These "signature varieties" provide both new tools to investigate paths and new challenging questions about their behavior. In this talk I will focus on the class of rough paths. They play a central role in stochastic analysis, and their signature variety has peculiar geometric properties, showing surprising analogies with the classical Veronese variety. We show that this so-called Rough Veronese is toric. This makes it much easier to study it and to perform explicit computations.