Functions on unparameterised path space
File(s)
Author(s)
William F., Turner
Type
Thesis or dissertation
Abstract
This thesis studies functions and topologies on unparameterised path space, obtained as the quotient of parameterised path space by tree-like equivalence. This space has gained prominence in recent years through its connection to the signature transform, a key tool in modern machine learning. We introduce and analyse candidate topologies for unparameterised path space in both bounded-variation and rough-path settings, focusing on separability, metrisability, and Polishness. We also investigate how the choice of topology
influences uniform approximation with linear functionals on the signature and the expected signature model.
Turning to functions on unparameterised path space, we examine two examples: scaling limits of randomised path developments and signature asymptotics. For the former, we generalise a recently established link between randomised general linear group path developments and the ordinary signature kernel to a broader class of randomisations. We also extend this framework to randomised unitary path developments, where connections with non-commutative probability emerge. For the latter we extend a result of Hambly & Lyons, who showed that the Hilbert–Schmidt norms of suitably rescaled signature terms recover a weighted curvature of the tree-reduced representative of each equivalence class. By exploiting connections between Wasserstein distances and expected fluctuations of Brownian bridges, we establish this result for a class of C2 curves improving on the original C3 assumption.
Finally, we propose LévyGAN: a generative modelling approach for producing approximate samples from the Lévy area of multi-dimensional Brownian motion, a quantity central to high-order SDE solvers. Our method leverages symmetries in the joint distribution of Brownian increments and Lévy area, removing the need for training data. We demonstrate its application through multilevel Monte Carlo simulation of the log-Heston model.
influences uniform approximation with linear functionals on the signature and the expected signature model.
Turning to functions on unparameterised path space, we examine two examples: scaling limits of randomised path developments and signature asymptotics. For the former, we generalise a recently established link between randomised general linear group path developments and the ordinary signature kernel to a broader class of randomisations. We also extend this framework to randomised unitary path developments, where connections with non-commutative probability emerge. For the latter we extend a result of Hambly & Lyons, who showed that the Hilbert–Schmidt norms of suitably rescaled signature terms recover a weighted curvature of the tree-reduced representative of each equivalence class. By exploiting connections between Wasserstein distances and expected fluctuations of Brownian bridges, we establish this result for a class of C2 curves improving on the original C3 assumption.
Finally, we propose LévyGAN: a generative modelling approach for producing approximate samples from the Lévy area of multi-dimensional Brownian motion, a quantity central to high-order SDE solvers. Our method leverages symmetries in the joint distribution of Brownian increments and Lévy area, removing the need for training data. We demonstrate its application through multilevel Monte Carlo simulation of the log-Heston model.
Version
Open Access
Date Issued
2025-09-28
Date Awarded
2026-01-01
Copyright Statement
Attribution-NonCommercial 4.0 International Licence (CC BY-NC)
License URL
Advisor
Thomas, Cass
Sponsor
Engineering and Physical Sciences Research Council
Publisher Department
Department of Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)