We prove two versions of a universal approximation theorems that allow to approximate continuous functions of càdlàg (rough) paths via linear functions 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 enable the inclusion of jumps. Indeed, as an application, we define a new class of signature models based on augmented Lévy processes, which we call Lévy-type signature models. They extend continuous signature models for asset prices as proposed e.g. by Perez-Arribas et al. in several directions, while still preserving universality and tractability properties. To analyze this, we first show that the signature process of a generic multivariate Lévy process is a polynomial process on the extended tensor algebra and then use this for pricing, hedging and calibration approaches within Lévy-type signature models.