Category Archives: Stochastic Calculus lectures

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 35. Weak differentiability for solutions of stochastic differential equations and the existence of a smooth density

As usual, we consider a filtered probability space which satisfies the usual conditions and on which is defined a -dimensional Brownian motion . Our purpose here, is to prove that solutions of stochastic differential equations are differentiable in the sense … Continue reading

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Lecture 34. The Wiener chaos expansion

As in the previous Lectures, we consider a filtered probability space on which is defined a Brownian motion , and we assume that is the usual completion of the natural filtration of . Our goal is here to write an … Continue reading

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Lecture 33. The Malliavin matrix and existence of densities

More generally, by using the same methods as in the previous Lecture, we can introduce iterated derivatives. If , we set . We may then consider as a square integrable random process indexed by and valued in . By using … Continue reading

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Lecture 32. The Malliavin derivative

The next Lectures will be devoted to the study of the problem of the existence of a density for solutions of stochastic differential equations. The basic tool to study such questions is the so-called Malliavin calculus. Let us consider a … Continue reading

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Lecture 31. Then, a miracle occurs

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Lecture 30. Stratonovitch stochastic differential equations

As usual, let be a filtered probability space which satisfies the usual conditions. It is often useful to use the language of Stratonovitch ‘s integration to study stochastic differential equations because the Itō’s formula takes a much nicer form. If … Continue reading

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