Two-parameter rough integration: theory and applications
Author(s)
Pei, Jeffrey
Type
Thesis
Abstract
In this thesis, we develop and expand on a two parameter rough integration theory based on the jointly controlled paths introduced by Hairer and Gerasimovičs in the paper "Hörmander’s theorem for semilinear SPDEs" (2019). We generalise the notion of jointly controlled paths to one for a larger class of driving rough paths, and use these jointly controlled paths as classes of twice rough integrable which satisfy a rough Fubini's theorem. One important example of a jointly controlled path is the signature kernel, the inner product of path signatures. Path signatures have seen increasing use in data science over the last decade, including use in kernel methods in more recent years.
For continuously differentiable paths, the signature kernel can be expressed as the solution of a Goursat problem, a type of linear second order hyperbolic PDE, which provides a method of computing signature kernels without working on the infinite dimensional spaces that path signatures lie on. Here we will use this representation of the signature kernel as the solution of a hyperbolic PDE to create an alternative approximation scheme for computation of the signature kernel through successive polynomial interpolation of boundary conditions.
For continuously differentiable paths, the signature kernel can be expressed as the solution of a Goursat problem, a type of linear second order hyperbolic PDE, which provides a method of computing signature kernels without working on the infinite dimensional spaces that path signatures lie on. Here we will use this representation of the signature kernel as the solution of a hyperbolic PDE to create an alternative approximation scheme for computation of the signature kernel through successive polynomial interpolation of boundary conditions.
Version
Open Access
Date Issued
2023-11-18
Date Awarded
01/07/2024
License URL
Advisor
Cass, Thomas
Sponsor
Engineering and Physical Sciences Research Council
Publisher Department
Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)