Prabhakar Lévy processes

We introduce here a generalization of the Mittag-Leffler L´evy process (with parameter $alpha$), obtained by extending its L´evy measure through the Prabhakar function (which is a Mittag-Leffler with the additional parameters $eta$ and $gamma$). We prove that this so-called Prabhakar process, in the special case $eta= 1$, can be represented as an $alpha$-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 $gamma = 1$. On the other hand, for $alpha = eta = 1$, 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.