## Abstract and Applied Analysis

### Alternative Forms of Compound Fractional Poisson Processes

#### Abstract

We study here different fractional versions of the compound Poisson process. The fractionality is introduced in the counting process representing the number of jumps as well as in the density of the jumps themselves. The corresponding distributions are obtained explicitly and proved to be solution of fractional equations of order less than one. Only in the final case treated in this paper, where the number of jumps is given by the fractional-difference Poisson process defined in Orsingher and Polito (2012), we have a fractional driving equation, with respect to the time argument, with order greater than one. Moreover, in this case, the compound Poisson process is Markovian and this is also true for the corresponding limiting process. All the processes considered here are proved to be compositions of continuous time random walks with stable processes (or inverse stable subordinators). These subordinating relationships hold, not only in the limit, but also in the finite domain. In some cases the densities satisfy master equations which are the fractional analogues of the well-known Kolmogorov one.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 747503, 30 pages.

Dates
First available in Project Euclid: 5 April 2013

https://projecteuclid.org/euclid.aaa/1365168348

Digital Object Identifier
doi:10.1155/2012/747503

Mathematical Reviews number (MathSciNet)
MR2991021

Zentralblatt MATH identifier
1255.60063

#### Citation

Beghin, Luisa; Macci, Claudio. Alternative Forms of Compound Fractional Poisson Processes. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 747503, 30 pages. doi:10.1155/2012/747503. https://projecteuclid.org/euclid.aaa/1365168348

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