Category Archives: Rough paths theory

Lecture 8. Rough paths Fall 2017

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

Posted in Rough paths theory | Leave a comment

Lecture 7. Rough paths. Fall 2017

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

Posted in Rough paths theory | 3 Comments

Lecture 6. Rough paths. Fall 2017

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

Posted in Rough paths theory | 2 Comments

Lecture 5. Rough paths. Fall 2017

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

Posted in Rough paths theory | 1 Comment

Lecture 4. Rough paths. Fall 2017

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

Posted in Rough paths theory | 3 Comments

Lecture 3 Rough paths. Fall 2017

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

Posted in Rough paths theory | Leave a comment

Rough paths theory Fall 2017. Lecture 2

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

Posted in Rough paths theory | Leave a comment