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2012 Alternative Forms of Compound Fractional Poisson Processes
Luisa Beghin, Claudio Macci
Abstr. Appl. Anal. 2012(SI01): 1-30 (2012). DOI: 10.1155/2012/747503


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.


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Luisa Beghin. Claudio Macci. "Alternative Forms of Compound Fractional Poisson Processes." Abstr. Appl. Anal. 2012 (SI01) 1 - 30, 2012.


Published: 2012
First available in Project Euclid: 5 April 2013

zbMATH: 1255.60063
MathSciNet: MR2991021
Digital Object Identifier: 10.1155/2012/747503

Rights: Copyright © 2012 Hindawi

Vol.2012 • No. SI01 • 2012
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