Hitting densities for spectrally positive stable processes

Abstract

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