Density behaviour related to Lévy processes
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- by Loïc Chaumont and Jacek Małecki PDF
- Trans. Amer. Math. Soc. 374 (2021), 1919-1945 Request permission
Abstract:
Let $p_t(x)$, $f_t(x)$ and $q_t^*(x)$ be the densities at time $t$ of a real Lévy process, its running supremum and the entrance law of the reflected excursions at the infimum. We provide relationships between the asymptotic behaviour of $p_t(x)$, $f_t(x)$ and $q_t^*(x)$, when $t$ is small and $x$ is large. Then for large $x$, these asymptotic behaviours are compared to this of the density of the Lévy measure. We show in particular that, under mild conditions, if $p_t(x)$ is comparable to $t\nu (x)$, as $t\rightarrow 0$ and $x\rightarrow \infty$, then so is $f_t(x)$.References
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Additional Information
- Loïc Chaumont
- Affiliation: LAREMA, Département de Mathématique, Université d’Angers, Bd Lavoisier - 49045, Angers Cedex 01, France
- Email: loic.chaumont@univ-angers.fr
- Jacek Małecki
- Affiliation: Wydział Matematyki, Politechnika Wrocławska, ul. Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland
- ORCID: 0000-0003-2250-5010
- Email: jacek.malecki@pwr.edu.pl
- Received by editor(s): January 23, 2020
- Received by editor(s) in revised form: June 11, 2020, June 23, 2020, and July 14, 2020
- Published electronically: December 18, 2020
- Additional Notes: The second author was supported by the Polish National Science Centre (NCN) grant no. 2015/19/B/ST1/01457 and the Wrocław University of Science and Technology grant 049U/0052/19
- © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 374 (2021), 1919-1945
- MSC (2020): Primary 60G51; Secondary 46N30
- DOI: https://doi.org/10.1090/tran/8268
- MathSciNet review: 4216728