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Smoothness of the law of some one-dimensional jumping SDEs with non-constant rate of jump

Abstract : We consider a one-dimensional jumping Markov process {X-t(x)}(t >= 0), solving a Poisson-driven stochastic differential equation. We prove that the law of X-t(x) admits a smooth density for t > 0, under some regularity and non-degeneracy assumptions on the coefficients of the S.D.E. To our knowledge, our result is the first one including the important case of a non-constant rate of jump. The main difficulty is that in such a case,the map x -> X-t(x) is not smooth. This seems to make impossible the use of Malliavin calculus techniques. To overcome this problem, we introduce a new method, in which the propagation of the smoothness of the density is obtained by analytic arguments.
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https://hal-upec-upem.archives-ouvertes.fr/hal-00693077
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Submitted on : Tuesday, May 1, 2012 - 7:58:33 PM
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  • HAL Id : hal-00693077, version 1

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Nicolas Fournier. Smoothness of the law of some one-dimensional jumping SDEs with non-constant rate of jump. Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2008, 13 (?). ⟨hal-00693077⟩

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