Abstract
In order to study the regularity of the density of a solution of a infinite activity jump driven stochastic differential equation we consider the following two-step approximation method. First, we use the solution of the moment problem in order to approximate the small jumps by another whose Lévy measure has finite support. In a second step we replace the approximation of the first two moments by a small noise Brownian motion based on the Assmussen-Rosiński approach. This approximation needs to satisfy certain properties in order to apply the “balance” method which allows the study of densities for the solution process based on Malliavin Calculus for the Brownian motion. Our results apply to situations where the Lévy measure is absolutely continuous with respect to the Lebesgue measure or purely atomic measures or combinations of them.
Funding Statement
The research of the third author was supported by KAKENHI grants 20K03666 20K03666 and 20K03666. This research is partly funded by the Bézout Labex, funded by ANR, reference ANR-10-LABX-58. L.C. also acknowledges the MIUR Excellence Department Project awarded to the Department of Mathematics of the University of Rome Tor Vergata and the Beyond Borders Project “Asymptotic Methods in Probability”.
Citation
Vlad Bally. Lucia Caramellino. Arturo Kohatsu-Higa. "Using moment approximations to study the density of jump driven SDEs." Electron. J. Probab. 27 1 - 21, 2022. https://doi.org/10.1214/22-EJP785
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