Alternative forms of compound fractional Poisson processes
type: Item
instance of: scholarly article
title: Alternative forms of compound fractional Poisson processes
zbMATH Open document ID: 1255.60063
DOI: 10.1155/2012/747503
author: Claudio Macci
author: L. Beghin
published in: Abstract and Applied Analysis
publication date: 2012-12-18T00:00:00Z
review text: Summary: We study here different fractional versions of the compound Poisson process. 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 by \textit{E. Orsingher} and \textit{F. Polito} [Stat. Probab. Lett. 82, No. 4, 852--858 (2012; Zbl 1270.60048)], 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.
Mathematics Subject Classification ID: 60G22
Mathematics Subject Classification ID: 60K05
zbMATH DE Number: 6116445
zbMATH Keywords: compound Poisson processes
zbMATH Keywords: fractional-difference Poisson processes
zbMATH Keywords: continuous-time random walks
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2012/747503
OpenAlex ID: W2096217566
Wikidata QID: Q58696717
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description: scientific article; zbMATH DE number 6116445