Rough Path Theory Spring 2021

Andrew Allan, Josef Teichmann


The lectures for this course are now finished. The lectures were recorded and the recordings may be found below.


The aim of this course is to provide an introduction to the theory of rough paths, with a particular focus on their integration theory and associated rough differential equations, and how the theory relates to and enhances the field of stochastic calculus.

Our first motivation will be to understand the limitations of classical notions of integration to handle paths of very low regularity, and to see how the rough integral succeeds where other notions fail. We will construct rough integrals and establish solutions of differential equations driven by rough paths, as well as the continuity of these objects with respect to the paths involved, and their consistency with stochastic integration and SDEs. Various applications and extensions of the theory will then be discussed.


The aim will be to make the course as self-contained as possible, but some knowledge of stochastic analysis is highly recommended. The course “Brownian Motion and Stochastic Calculus” would be ideal, but not strictly required.

Lecture Notes


Exercise sheets will be posted here. The exercises are provided to give you some practice with the material, but students are not expected to hand in their work. There will be opportunities to ask questions about the exercises during the lectures.

Exercise sheet Solutions
Exercise sheet 1 Solution sheet 1
Exercise sheet 2 Solution sheet 2
Exercise sheet 3 Solution sheet 3

Lecture recordings

Topic of lecture Recording
Introduction Lecture 1
Hölder spaces and the space of rough paths Lecture 2
Brownian motion as a rough path Lecture 3
The sewing lemma and Young integration Lecture 4
More Young integration and controlled paths Lecture 5
Rough integration Lecture 6
Itô's formula for rough paths Lecture 7
Rough differential equations Lecture 8
Consistency with stochastic calculus Lecture 9
Pathwise stability of likelihood estimators Lecture 10
An introduction to regularity structures - Part 1 (by Josef Teichmann) Lecture 11
An introduction to regularity structures - Part 2 (by Josef Teichmann) Lecture 12
Parameter uncertainty in stochastic filtering Lecture 13
The stochastic sewing lemma Lecture 14