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2014 Geometric stable processes and related fractional differential equations
Luisa Beghin
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Electron. Commun. Probab. 19: 1-14 (2014). DOI: 10.1214/ECP.v19-2771


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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Luisa Beghin. "Geometric stable processes and related fractional differential equations." Electron. Commun. Probab. 19 1 - 14, 2014.


Accepted: 1 March 2014; Published: 2014
First available in Project Euclid: 7 June 2016

zbMATH: 1321.60101
MathSciNet: MR3174831
Digital Object Identifier: 10.1214/ECP.v19-2771

Primary: 60G52
Secondary: 26A33 , 33E12 , 34A08

Keywords: Gamma subordinator , Geometric Stable subordinator , Riesz-Feller fractional derivative , ‎shift operator , Symmetric Geometric Stable law


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