Introduction to rough paths          

Vlad Bally
Université Gustave Eiffel

A course for the PhD School in Mathematics at Tor Vergata
Spring semester 2022

Rough path theory has been initiated in the last 90's by Terry Lyons in [1]. Then, in the last 20 years it has had a tremendous development, including the "regularity structures" theory of Hairer (we will not touch to this last topic in our course). And nowadays this is still an extremely active area of research. The aim of this theory is to construct a variant of the stochastic integral which is "pathwise". Moreover one solves Stochastic Differential Equations (SDEs) with the usual stochastic integral replaced by the "rough integral". This gives an application defined on the space of continuous functions C([0,T]) which, under the Wiener measure, produced the solution of the SDE. We stress that the classical theory of stochastic flows (due to Kunita, Bismut, and many others) produces a "strong solution" of the SDE, which is exactly such an application. But there is a crucial progress here: in the classical case, the flow produces a solution "almost surely" with an exception set depending on the coefficients of the SDE, whereas in the rough path theory the exception set is independent of the coefficients (in some sense it is universal). Moreover, a continuity property of the application, with respect to a specific norm (the "rough path norm") is proved.
Nowadays there are many text books devoted to this subject. They are more or less difficult to read because of a rather heavy technical background. The aim of this introductory course is to give an elementary and simple approach to the main ideas in this theory. But of course, this is just a first step and a deep knowledge of the theory needs to read one of these books. I strongly recommend the beautiful book [2] of Friz and Hairer (which I will more or less follow).

Lecture notes: click here.

[1] Terry J. Lyons. Differential equations driven by rough signals. Rev. Mat. Iberoamericana, 14(2), 215-310, 1998.
[2] Peter K. Friz and Martin Hairer. A Course on Rough Paths with an introduction to regularity structures. Volume XIV of Universitext. Springer, Berlin, 2014.
[3] Denis Feyel and Arnaud de La Pradelle. Curvilinear integrals along enriched paths. Electronic Journal of Probability, 2006.
[4] Peter K. Friz and Nicolas B. Victoir. Multidimensional stochastic processes as rough paths. Volume 120 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2010. Theory and applications.

Schedule: the course is held in blended form (in presence and online) from May 4, 2022, to May 26, 2022, in room 1201 (aula Dal Passo), from 2pm to 4pm on the following days:

The lessons will be streamed via Microsoft Teams. Those interested can request to be included in the Teams classroom of the course or to receive the link to follow the lessons by writing an email to Lucia Caramellino.

For further information: please write an email to Vlad Bally or to Lucia Caramellino (local organizer).