Geometric Stable processes and related fractional differential equations

We are interested in the differential equations satisfied by the density of the Geometric Stable processes $\mathcal{G}_α^β=\left\{\mathcal{G}_α^β(t);t\geq 0\right\}$, with stability \ index $$ and asymmetry parameter $β\in \lbrack -1,1]$, both in the univariate and in the multivariate cases. We resort to their representation as compositions of stable processes with an independent Gamma subordinator. As a preliminary result, we prove that the latter is governed by a differential equation expressed by means of the shift operator. As a consequence, we obtain the space-fractional equation satisfied by the density of $\mathcal{G}_α^β.$ For some particular values of $$ 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.

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