Alternative Forms of Compound Fractional Poisson Processes
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 o...
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Wiley
2012-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2012/747503 |
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| author | Luisa Beghin Claudio Macci |
| author_facet | Luisa Beghin Claudio Macci |
| author_sort | Luisa Beghin |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-512c6da4c9b442ef98474baf14403ccd |
| institution | OA Journals |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-512c6da4c9b442ef98474baf14403ccd2025-08-20T02:05:17ZengWileyAbstract and Applied Analysis1085-33751687-04092012-01-01201210.1155/2012/747503747503Alternative Forms of Compound Fractional Poisson ProcessesLuisa Beghin0Claudio Macci1Dipartimento di Scienze Statistiche, Sapienza Università di Roma, Piazzale Aldo Moro 5, 00185 Roma, ItalyDipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 00133 Roma, ItalyWe 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.http://dx.doi.org/10.1155/2012/747503 |
| spellingShingle | Luisa Beghin Claudio Macci Alternative Forms of Compound Fractional Poisson Processes Abstract and Applied Analysis |
| title | Alternative Forms of Compound Fractional Poisson Processes |
| title_full | Alternative Forms of Compound Fractional Poisson Processes |
| title_fullStr | Alternative Forms of Compound Fractional Poisson Processes |
| title_full_unstemmed | Alternative Forms of Compound Fractional Poisson Processes |
| title_short | Alternative Forms of Compound Fractional Poisson Processes |
| title_sort | alternative forms of compound fractional poisson processes |
| url | http://dx.doi.org/10.1155/2012/747503 |
| work_keys_str_mv | AT luisabeghin alternativeformsofcompoundfractionalpoissonprocesses AT claudiomacci alternativeformsofcompoundfractionalpoissonprocesses |