State-Dependent Fractional Point Processes

In this paper we analyse the fractional Poisson process where the state probabilities p k ν k (t), t ≥ 0, are governed by time-fractional equations of order 0 < ν k ≤ 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 p...

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Bibliographic Details
Published in:Journal of applied probability 2015-03, Vol.52 (1), p.18-36
Main Authors: Garra, R., Orsingher, E., Polito, F.
Format: Article
Language:English
Subjects:
26A33
34A08
60G22
60G55
Dzhrbashyan-Caputo fractional derivative
Fractions
Laplace transforms
Mittag-Leffler function
Poisson distribution
Poisson process
pure birth process
stable process
Studies
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Summary:In this paper we analyse the fractional Poisson process where the state probabilities p k ν k (t), t ≥ 0, are governed by time-fractional equations of order 0 < ν k ≤ 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 p k ν k (t) and various representations of state probabilities. We show that the Poisson process with intermediate waiting times depending on ν k differs from that constructed from the fractional state equations (in the case of ν k = 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.
ISSN:0021-9002
1475-6072
DOI:10.1239/jap/1429282604