Upper bounds for the derivatives of the density associated to solutions of stochastic differential equations with jumps

In this article, we prove the existence and regularity of densities associated to stochastic differential equations driven by a jump process with infinite activity. Furthermore, we provide explicit bounds for the density and its derivatives. The main argument is based on two ideas: (1) The use of th...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2024-03, Vol.531 (2), p.127817, Article 127817
Main Authors: Bally, V., Caramellino, L., Kohatsu-Higa, A.
Format: Article
Language:English
Subjects:
Asmussen-Rosińki approximation
Densities
Jump diffusion semigroup
Regularity
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Summary:In this article, we prove the existence and regularity of densities associated to stochastic differential equations driven by a jump process with infinite activity. Furthermore, we provide explicit bounds for the density and its derivatives. The main argument is based on two ideas: (1) The use of the regularity of the Brownian motion approximation of small jumps of an infinite activity process (the so-called Asmussen-Rosiński approximation). (2) The improvement of the approximation procedure using a Taylor like expansion of the law based on the so-called Levi parametrix method.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2023.127817