Hitting densities for spectrally positive stable processes

TL;DR

This paper establishes a multiplicative identity for hitting times of spectrally positive stable processes, enabling explicit density calculations, fractional moments, and demonstrating unimodality for certain stability indices.

Contribution

It introduces a new multiplicative identity in law for hitting times of spectrally positive stable processes, simplifying density and moment computations and proving unimodality for specific cases.

Findings

Derived series representations for densities.

Computed fractional moments explicitly.

Proved unimodality for ppa  3/2.

Abstract

A multiplicative identity in law connecting the hitting times of completely asymmetric $\alpha-$stable L\'evy 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 $\alpha\le 3/2.$ Analogous results are obtained, in a much simplified manner, for the first passage time across a positive level.