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Peer-reviewedCC BYJournal of Mathematical Analysis and Applications

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

  • V. Bally
  • L. Caramellino
  • A. Kohatsu-Higa
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Abstract

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.

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Cite

BibTeXRIS
BibTeX
@article{bally2026upper,
  title = {{Upper bounds for the derivatives of the density associated to solutions of stochastic differential equations with jumps}},
  author = {V. Bally and L. Caramellino and A. Kohatsu-Higa},
  journal = {Journal of Mathematical Analysis and Applications},
  year = {2026},
  month = nov,
  doi = {10.1016/j.jmaa.2023.127817},
  url = {https://hal.science/hal-05854675v1},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - Upper bounds for the derivatives of the density associated to solutions of stochastic differential equations with jumps
AU  - V. Bally
AU  - L. Caramellino
AU  - A. Kohatsu-Higa
PY  - 2026
DA  - 2026/11/06
JO  - Journal of Mathematical Analysis and Applications
DO  - 10.1016/j.jmaa.2023.127817
UR  - https://hal.science/hal-05854675v1
AB  - 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.
SN  - 0022-247X
SN  - 1096-0813
ER  -