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Category Archives: Rough paths theory
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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Rough paths theory Fall 2017. Lecture 1
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
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Rough paths theory. Fall 2017
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
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Diffusion processes and stochastic calculus textbook
My book which is published by the European Mathematical Society is now available. Diffusion Processes and Stochastic Calculus The content is partially based on the lecture notes in stochastic calculus and rough paths theory which are posted on this blog … Continue reading