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Geometric stable processes and related fractional differential equations


 
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1. Title Title of document Geometric stable processes and related fractional differential equations
 
2. Creator Author's name, affiliation, country Luisa Beghin; Sapienza University of Rome; Italy
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Symmetric Geometric Stable law; Geometric Stable subordinator; Shift operator; Riesz-Feller fractional derivative; Gamma subordinator.
 
3. Subject Subject classification 60G52; 34A08; 33E12; 26A33.
 
4. Description 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.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s)
 
7. Date (YYYY-MM-DD) 2014-03-01
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ecp.ejpecp.org/article/view/2771
 
10. Identifier Digital Object Identifier 10.1214/ECP.v19-2771
 
11. Source Journal/conference title; vol., no. (year) Electronic Communications in Probability; Vol 19
 
12. Language English=en en
 
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