Electronic Communications in Probability

Geometric stable processes and related fractional differential equations

Luisa Beghin

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G52: Stable processes
Secondary: 34A08: Fractional differential equations 33E12: Mittag-Leffler functions and generalizations 26A33: Fractional derivatives and integrals

Symmetric Geometric Stable law Geometric Stable subordinator Shift operator Riesz-Feller fractional derivative Gamma subordinator

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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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