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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
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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
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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
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Lecture 5. Rough paths. Fall 2017
In this lecture we define the Young‘s integral when and with . The cornerstone is the following YoungLoeve estimate. Theorem: Let and . Consider now with . The following estimate holds: for , Proof: For , let us define We … Continue reading
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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
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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
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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 1variation topology of the solution of an equation with respect to the driving signal. Theorem: … Continue reading
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