Smoothness of the law of some one-dimensional jumping S.D.E.s with non-constant rate of jump
Nicolas Fournier, Université Paris Est
Abstract
We consider a one-dimensional
jumping Markov process,
solving a Poisson-driven stochastic differential equation.
We prove that the law of this process admits a smooth density for all
positive times,
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 process is not smooth
as a function of its initial condition.
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.
Full text: PDF | PostScript
Copyright for articles published in this journal is retained by the authors, with first publication rights granted to the journal. By virtue of their appearance in this open access journal, articles are free to use, with proper attribution, in educational and other non-commercial settings.
The authors of papers published in EJP/ECP retain the copyright. We ask for the permission to use the material in any form. We also require that the initial publication in EJP or ECP is acknowledged in any future publication of the same article.
Before a paper is published in the Electronic Journal of Probability or Electronic Communications in Probability we must receive a hard-copy of the copyright form. Please mail it to
Philippe Carmona
Laboratoire Jean Leray UMR 6629
Universite de Nantes,
2, Rue de la Houssinière BP 92208
F-44322 Nantes Cédex 03
France
You can also send it by FAX: (33|0) 2 51 12 59 12 to the attention of Philippe Carmona. You can even send a scanned jpeg or pdf of this copyright form to the managing editor ejpecpme@math.univ-nantes.fr. as an attached file.
If a paper has several authors, the corresponding author signs the copyright form on behalf of all the authors.