Hitting densities for spectrally positive stable processes

Abstract

A multiplicative identity in law connecting the hitting times of completely asymmetric α-stable Lévy processes in duality is established. In the spectrally positive case, this identity allows with an elementary argument to compute fractional moments and to get series representations for the density. We also prove that the hitting times are unimodal as soon as . Analogous results are obtained for the first passage time across a positive level, in a simple manner.

Keywords:

• exit time
• hitting time
• Mellin transform
• series representation
• stable Lévy processes
• unimodality

2000 Mathematics Subject Classification::

• 60E05
• 60G52

Acknowledgements

Part of this work was done during a sunny stay at the University of Tokyo, and I am grateful to N. Yoshida for his hospitality. Ce travail a aussi bénéficié d'une aide de l'Agence Nationale de la Recherche portant la référence ANR-09-BLAN-0084-01.