Category Archives: Rough paths theory

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

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Lecture notes on rough paths theory

For those who are interested, here are the notes corresponding to the lectures posted on this blog. All comments are welcome.

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Lecture 30. The Stroock-Varadhan support theorem

To conclude this course, we are going to provide an elementary proof of the Stroock–Varadhan support theorem which is based on rough paths theory. We first remind that the support of a random variable which defined on a metric space … Continue reading

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Lecture 29. Stochastic differential equations as rough differential equations

Based on the results of the previous Lecture, it should come as no surprise that differential equations driven by the Brownian rough path should correspond to Stratonovitch differential equations. In this Lecture, we prove that it is indeed the case. … Continue reading

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Lecture 28. Signature of the Brownian rough path

Since a -dimensional Brownian motion is a -rough path for , we know how to give a sense to the signature of the Brownian motion. In particular, the iterated integrals at any order of the Brownian motion are well defined … Continue reading

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Lecture 27. Approximation of the Brownian rough path

Our goal in the next two lectures will be to prove that rough differential equations driven by a Brownian motion seen as a -rough path, are nothing else but stochastic differential equations understood in the Stratonovitch sense. The proof of … Continue reading

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Lecture 26. Lyons’ continuity theorem: Proof

We now turn to the proof of Lyons’ continuity theorem. Theorem: Let . Assume that are -Lipschitz vector fields in . Let such that with . Let be the solutions of the equations There exists a constant depending only on … Continue reading

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