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State-Dependent Fractional Point Processes

Part of: Real functions General theory in ordinary differential equations 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: [email protected]
∗∗ 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.
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Abstract

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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 

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