Universality of Signatures in Rough Path Spaces

Universality of Signatures in Rough Path Spaces: A Kernel-Theoretic Approach to Local and Global Approximations

Carrondo do Amaral Magalhães Simões, Tomás (TU Delft Electrical Engineering, Mathematics and Computer Science; TU Delft Delft Institute of Applied Mathematics)

Yu, F. (mentor)
Cuchiero, Christa (mentor)
Redig, F.H.J. (graduation committee)
Bartolucci, F. (graduation committee)

Delft University of Technology

Applied Mathematics

2024-06-21

This thesis examines the approximation capabilities of path signatures within rough path spaces, focusing on both local and global universality. To this end, we provide a self-contained introduction to Rough Path theory, highlighting the interplay between additive and multiplicative functionals. This leads to the renowned Lyons' Extension theorem and the definition of rough path spaces. We also re-examine the concept of universality from a kernel-theoretic perspective, culminating in the classical universal approximation result for signatures over a compact subset of paths. To broaden the scope beyond compact domains, we introduce the framework of weighted spaces and elaborate on the notion of global universality. Specifically, we formally define globally universal kernels and prove sufficient conditions for their existence. The associated reproducing kernel Hilbert space is shown to approximate a wide range of functions over the entire domain, which may be non-(locally) compact. In particular, we apply these theoretical tools to rough path spaces, thereby cementing the global universality of signatures.

Signatures
Rough Path Theory
Kernels
Universality
Reproducing Kernel Hilbert Spaces

http://resolver.tudelft.nl/uuid:52c1fd89-f57b-4937-bab5-5781c395e152

Student theses

master thesis

© 2024 Tomás Carrondo do Amaral Magalhães Simões