## Electronic Communications in Probability

### Geometric stable processes and related fractional differential equations

Luisa Beghin

#### 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.

#### Article information

Source
Electron. Commun. Probab., Volume 19 (2014), paper no. 13, 14 pp.

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

https://projecteuclid.org/euclid.ecp/1465316715

Digital Object Identifier
doi:10.1214/ECP.v19-2771

Mathematical Reviews number (MathSciNet)
MR3174831

Zentralblatt MATH identifier
1321.60101

Rights

#### Citation

Beghin, Luisa. Geometric stable processes and related fractional differential equations. Electron. Commun. Probab. 19 (2014), paper no. 13, 14 pp. doi:10.1214/ECP.v19-2771. https://projecteuclid.org/euclid.ecp/1465316715

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