March 2015 State-dependent fractional point processes
R. Garra, E. Orsingher, F. Polito
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J. Appl. Probab. 52(1): 18-36 (March 2015). DOI: 10.1239/jap/1429282604


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.


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R. Garra. E. Orsingher. F. Polito. "State-dependent fractional point processes." J. Appl. Probab. 52 (1) 18 - 36, March 2015.


Published: March 2015
First available in Project Euclid: 17 April 2015

zbMATH: 1315.60056
MathSciNet: MR3336844
Digital Object Identifier: 10.1239/jap/1429282604

Primary: 26A33 , 60G55
Secondary: 34A08 , 60G22

Keywords: Dzhrbashyan-Caputo fractional derivative , Mittag-Leffler function , Poisson process , pure birth process , Stable process

Rights: Copyright © 2015 Applied Probability Trust


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Vol.52 • No. 1 • March 2015
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