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

21 Sep 2022

Rough Path Interest Group

Abstract

Signature-based models have recently entered the field of stochastic modeling, in particular in Mathematical Finance. The choice of the signature as main building block is mostly explained by a universal approximation theorem (UAT) according to which continuous functions of continuous paths can be approximated by linear functionals of the time-extended signature. This powerful result however, leaves open the question of approximating continuous functions of the more general set of càdlàg paths. Based on recent advances on the signature of càdlàg paths and by using appropriate topologies thereon, during the first part of the talk we present two versions of a UAT that solve this question. Then, as an important application, we define a new class of signature models based on an augmented Lévy process, which we call Lévy-type signature models. They extend the class of continuous signature models for asset prices proposed so far 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 and hedging approaches within Lévy-type signature models. This presentation is based on a on joint work with Christa Cuchiero and Sara Svaluto-Ferro.

Our speaker

Francesca is a second-year PhD student in Mathematics at the University of Vienna, supervised by Professor Christa Cuchiero. In 2020 she graduated with an MSc from the University of Turin and obtained a Master in Mathematics for finance and data from Université Paris-Est Marne-la-Vallée. Her research interests include signature methods and their applications to data science and mathematical finance.

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