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Theory of Probability and Mathematical Statistics

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Malliavin calculus for Lévy processes with arbitrary Lévy measures

Author: A. M. Kulik
Translated by: Oleg Klesov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 72 (2005).
Journal: Theor. Probability and Math. Statist. 72 (2006), 75-92
MSC (2000): Primary 60H07; Secondary 60G51
Published electronically: August 18, 2006
MathSciNet review: 2168138
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Abstract: A new method is proposed to prove the absolute continuity of distributions of solutions of stochastic differential equations with jumps. The method is based on the differentiation in time in the space of functionals of the Poisson point measure. In contrast to the known Bismut and Picard methods, our approach can be applied to point measures with arbitrary Lévy measures. We obtain sufficient conditions for the absolute continuity of the solutions expressed in terms of the coefficients of the equation; the conditions do not involve assumptions on properties of the Lévy measure.

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Additional Information

A. M. Kulik
Affiliation: Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkovskaya 3, Kiev 01601, Ukraine

Keywords: Malliavin calculus, L\'evy process, stratification method, admissible time-stretching transformations
Received by editor(s): March 19, 2004
Published electronically: August 18, 2006
Additional Notes: Supported by the Ministry of Science and Education of Ukraine, Project # 01.07/103.
Article copyright: © Copyright 2006 American Mathematical Society