Prabhakar Lévy processes

Abstract

We introduce here a generalization of the Mittag-Leffler Lévy process (with parameter α), obtained by extending its Lévy measure through the Prabhakar function (which is a Mittag-Leffler with the additional parameters β and γ). We prove that this so-called Prabhakar process, in the special case β=1, can be represented as an α-stable process subordinated by an independent generalized gamma subordinator; thus it can be considered as an extension of the geometric stable process, to which it reduces for γ=1. On the other hand, for α=β=1, it coincides with the generalized gamma process itself. Therefore, by suitably specifying the three parameters, the Prabhakar process turns out to represent an interpolation among various well-known and widely applied stochastic models.