Geometric Stable processes and related fractional differential equations

Abstract

We are interested in the differential equations satisfied by the density of the Geometric Stable processes ${\mathcal{G}}_{\alpha }^{\beta }=\left\{{\mathcal{G}}_{\alpha }^{\beta }(t);t\ge 0\right\}$, with stability \ index and asymmetry parameter $\beta \in [-1,1]$, both in the univariate and in the multivariate cases. We resort to their representation as compositions of stable processes with an independent Gamma subordinator. As a preliminary result, we prove that the latter is governed by a differential equation expressed by means of the shift operator. As a consequence, we obtain the space-fractional equation satisfied by the density of ${\mathcal{G}}_{\alpha }^{\beta }.$ For some particular values of and $\beta ,$ we get some interesting results linked to well-known processes, such as the Variance Gamma process and the first passage time of the Brownian motion.

12 pages