Category Archives: Rough paths theory

In this lecture, it is now time to harvest the fruits of the two previous lectures. This will allow us to finally define the notion of -rough path and to construct the signature of such path. A first result which … Continue reading →

In the previous lecture we introduced the signature of a bounded variation path as the formal series If now , the iterated integrals can only be defined as Young integrals when . In this lecture, we are going to derive … Continue reading →

In this lecture we introduce the central notion of the signature of a path which is a convenient way to encode all the algebraic information on the path which is relevant to study differential equations driven by . The motivation … Continue reading →

In this lecture we define the Young‘s integral when and with . The cornerstone is the following Young-Loeve estimate. Theorem: Let and . Consider now with . The following estimate holds: for , Proof: For , let us define We … Continue reading →

Our next goal in this course is to define an integral that can be used to integrate rougher paths than bounded variation. As we are going to see, Young’s integration theory allows to define as soon as has finite -variation … Continue reading →

Let and let be a Lipschitz continuous map. In order to analyse the solution of the differential equation, and make the geometry enter into the scene, it is convenient to see as a collection of vector fields , where the … Continue reading →

In this lecture we establish the basic existence and uniqueness results concerning differential equations driven by bounded variation paths and prove the continuity in the 1-variation topology of the solution of an equation with respect to the driving signal. Theorem: … Continue reading →

The first few lectures are essentially reminders of undergraduate real analysis materials. We will cover some aspects of the theory of differential equations driven by continuous paths with bounded variation. The point is to fix some notations that will be … Continue reading →

During the Fall 2017, I will be teaching rough paths theory at the University of Connecticut. The course will be mainly based on those   notes and the lectures already posted on this blog in 2013 (when I first taught the class … Continue reading →