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

We prove a universal approximation theorem that allows approximating continuous functionals of càdlàg (rough) paths uniformly in time and on compact sets of paths via linear functionals of their time-extended signature. Our main motivation to treat this question comes from signature-based models for finance that allow the inclusion of jumps. Indeed, as an important application, we define a new class of universal signature models based on an augmented Lévy process, which we call Lévy-type signature models. They extend continuous signature models for asset prices as proposed e.g. by Perez Arribas et al. (Proceedings of the First ACM International Conference on AI in Finance, ICAIF’20, Association for Computing Machinery, New York, 1–8, 2021) in several directions, while still preserving universality and tractability properties. To analyse 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 and hedging approaches within Lévy-type signature models.