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 {Gβα(t); t≥0}, with stability index α ∈ (0,2] and symmetry parameter β ∈ [-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 transition density of Gβα(t). For some particular values of α and β, 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.