  We prove a universal approximation theorem that allows to approximate
continuous functionals of c\`adl\`ag (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 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.

Probability;Mathematical Finance