Universal approximation theorems for continuous functions of càdlàg paths and Lévy-type signature models

We prove two versions of a universal approximation theorem that allow to approximate continuous functions of ca`dla`g (rough) paths via linear functionals of their time-extended signature, one with respect to the Skorokhod J1-topology and the other one with respect to (a rough path version of) the Skorokhod M1-topology. Our main motivation to treat this question comes from signature-based models for finance that allow for the inclusion of jumps. Indeed, as an important application, we define a new class of universal signature models based on an augmented L ́evy process, which we call L ́evy-type signature models. They extend continuous signature models for asset prices as proposed e.g. by Arribas et al. (2020) in several directions, while still preserving universality and tractability properties. To analyze this, we first show that the signature process of a generic multivariate L ́evy process is a polynomial process on the extended tensor algebra and then use this for pricing and hedging approaches within L ́evy-type signature models.