Hostname: page-component-8448b6f56d-dnltx Total loading time: 0 Render date: 2024-04-24T08:33:35.247Z Has data issue: false hasContentIssue false
  • English
  • Français

State-Dependent Fractional Point Processes

Part of: General theory in ordinary differential equations Real functions Stochastic processes

Published online by Cambridge University Press:  30 January 2018

R. Garra ,
E. Orsingher  and
F. Polito
R. Garra*
Affiliation:
Sapienza Università di Roma
E. Orsingher*
Affiliation:
Sapienza Università di Roma
F. Polito*
Affiliation:
Università degli Studi di Torino
*
∗ Postal address: Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza Università di Roma, Via A.Scarpa 16, 00161, Roma, Italy. Email address: rolinipame@yahoo.it
∗∗ Postal address: Dipartimento di Statistica, Sapienza Università di Roma, Piazzale Aldo Moro 5, 00185 Roma, Italy.
∗∗∗ Postal address: Dipartimento di Matematica ‘G. Peano’, Università degli Studi di Torino, Via Carlo Alberto 10, 10123 Torino, Italy.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we analyse the fractional Poisson process where the state probabilities pkνk(t), t ≥ 0, are governed by time-fractional equations of order 0 < νk ≤ 1 depending on the number k of events that have occurred up to time t. We are able to obtain explicitly the Laplace transform of pkνk(t) and various representations of state probabilities. We show that the Poisson process with intermediate waiting times depending on νk differs from that constructed from the fractional state equations (in the case of νk = ν, for all k, they coincide with the time-fractional Poisson process). We also introduce a different form of fractional state-dependent Poisson process as a weighted sum of homogeneous Poisson processes. Finally, we consider the fractional birth process governed by equations with state-dependent fractionality.

Keywords

Dzhrbashyan-Caputo fractional derivativePoisson processstable processMittag-Leffler functionpure birth process

MSC classification

Primary: 60G55: Point processes 26A33: Fractional derivatives and integrals
Secondary: 34A08: Fractional differential equations 60G22: Fractional processes, including fractional Brownian motion
Type
Research Article
Information
Journal of Applied Probability , Volume 52 , Issue 1 , March 2015 , pp. 18 - 36
Copyright
© Applied Probability Trust 

References

Balakrishnan, N. and Kozubowski, T. J. (2008). A class of weighted Poisson processes. Statist. Prob. Lett. 78, 2346–2352.Google Scholar
Beghin, L. and Macci, C. (2013). Large deviations for fractional Poisson processes. Statist. Prob. Lett. 83, 1193–1202.CrossRefGoogle Scholar
Beghin, L. and Orsingher, E. (2009). Fractional Poisson processes and related planar random motions. Electron. J. Prob. 14, 1790–1827.CrossRefGoogle Scholar
D'Ovidio, M., Orsingher, E. and Toaldo, B. (2014). Fractional telegraph-type equations and hyperbolic Brownian motion. Statist. Prob. Lett. 89, 131–137.Google Scholar
Fedotov, S., Ivanov, A. O. and Zubarev, A. Y. (2013). Non-homogeneous random walks, subdiffusive migration of cells and anomolous chemotaxis. Math. Model. Nat. Phenom. 8, 28–43.Google Scholar
Garra, R. and Polito, F. (2011). A note on fractional linear pure birth and pure death processes in epidemic models. Physica A 390, 3704–3709.Google Scholar
Hilfer, R. and Anton, L. (1995). Fractional master equation and fractal time random walks. Phys. Rev. E 51, R848–R851.CrossRefGoogle ScholarPubMed
Laskin, N. (2003). Fractional Poisson process. Commun. Nonlinear Sci. Numerical Simul. 8, 201–213.Google Scholar
Laskin, N. (2009). Some applications of the fractional Poisson probability distribution. J. Math. Phys. 50, 113513.CrossRefGoogle Scholar
Mainardi, F., Gorenflo, R. and Scalas, E. (2004). A fractional generalization of the Poisson process. Vietnam J. Math. 32, 53–64.Google Scholar
Mathai, A. M. and Haubold, H. J. (2008). Special Functions for Applied Scientists. Springer, New York.CrossRefGoogle Scholar
Meerschaert, M. M., Nane, E. and Vellaisamy, P. (2011). The fractional poisson process and the inverse stable subordinator. Electron. J. Prob. 16, 1600–1620.Google Scholar
Olver, F. W. J., Lozier, D. W., Boisvert, R. F. and Clark, C. W. (eds) (2010). NIST Handbook of Mathematical Functions. Cambridge University Press.Google Scholar
Orsingher, E. and Polito, F. (2010). Fractional pure birth processes. Bernoulli 16, 858–881.Google Scholar
Orsingher, E. and Polito, F. (2012). The space-fractional Poisson process. Statist. Prob. Lett. 82, 852–858.Google Scholar
Podlubny, I. (1999). Fractional Differential Equations. Academic Press, San Diego, CA.Google Scholar
Prabhakar, T. R. (1971). A singular integral equation with a generalized Mittag–Leffler function in the kernel. Yokohama Math. J. 19, 7–15.Google Scholar
Repin, O. N. and Saichev, A. I. (2000). Fractional Poisson law. Radiophysics Quantum Electron. 43, 738–741.Google Scholar
Saxena, R. K., Mathai, A. M. and Haubold, H. J. (2006). Solutions of fractional reaction-diffusion equations in terms of Mittag–Leffler functions. Internat. J. Scientific Res. 15, 1–17.Google Scholar
Sixdeniers, J.-M., Penson, K. A. and Solomon, A. I. (1999). Mittag–Leffler coherent states. J. Phys. A 32, 7543–7563.Google Scholar