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\geq 0\right} $, with stability \ index $% \alpha \in (0,2]$ and asymmetry parameter $\beta \in \lbrack -1,1]$, 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 $% \alpha $ 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.