Geometric stable processes and related fractional differential equations

Luisa Beghin (Sapienza University of Rome)


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 symmetry 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.

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Pages: 1-14

Publication Date: March 1, 2014

DOI: 10.1214/ECP.v19-2771


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