@article{ECP2771,
author = {Luisa Beghin},
title = {Geometric stable processes and related fractional differential equations},
journal = {Electron. Commun. Probab.},
fjournal = {Electronic Communications in Probability},
volume = {19},
year = {2014},
keywords = {Symmetric Geometric Stable law; Geometric Stable subordinator; Shift operator; Riesz-Feller fractional derivative; Gamma subordinator.},
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 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.},
pages = {no. 13, 1-14},
issn = {1083-589X},
doi = {10.1214/ECP.v19-2771},
url = {http://ecp.ejpecp.org/article/view/2771}}