Description

The stochastic Itô integral allows for the integration of stochastic processes with highly oscillatory (i.e. "rough") trajectories, and, with countless applications from finance and biology to engineering and aerospace, allows for a stochastic calculus which has over recent decades provided the backbone for the mathematical analysis of systems under the influence of random noise. However, stochastic integration is crucially dependent on the specification of a probabilistic model, which in practice is almost always unknown. Moreover, the construction of stochastic integrals is inherently probabilistic, and it does not tell us how to integrate individual realisations of stochastic processes.

The theory of rough paths has emerged as novel approach for dealing with interactions in complex random systems. In particular, it provides a framework for the study of differential equations driven by highly irregular signals, without relying on probabilistic arguments. The key insight was to identify the precise characteristics of a path which are required for a purely analytical (i.e. non-probabilistic) notion of integration. A path enhanced with these characteristics is known as a "rough path", and differential equations driven by such paths are known as "rough differential equations".

Initiated by Terry Lyons in 1998, rough path theory has quickly become a cornerstone of modern stochastic analysis, and is an increasingly active research topic, with applications across many fields, including stochastic analysis, statistics, mathematical finance and data science.

This project would begin with a gentle review of some of the basic ideas and theory of rough paths. There would then be some flexibility regarding the direction in which you may wish to take the project. Possible topics include rough PDEs, rough paths with jumps, hybrid rough-stochastic equations, or applications of rough path techniques to robust statistics or to machine learning.

Prerequisites

The main requirement is a reasonable knowledge and motivation for analysis and probability. Some prior knowledge of stochastic processes and Itô calculus would be particularly helpful. Either Mathematical Finance III or Stochastic Processes III would be suitable prerequisites, and Analysis III may also be helpful.

Resources

By searching the internet you can find various sets of lecture notes on rough paths. For instance, you might like to read the introduction of these notes to get an idea of what the subject is about:

• A. L. Allan, Lecture notes on rough path theory, 2021.

There are also a number of books on the subject, for instance:

• P. K. Friz and M. Hairer, A course on rough paths with an introduction to regularity structures, 2nd ed., Springer, 2020.
• P. K. Friz and N. B. Victoir, Multidimensional stochastic processes as rough paths, vol. 120 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2010.