Journal of Applied Probability

State-dependent fractional point processes

R. Garra, E. Orsingher, and F. Polito

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Abstract

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.

Article information

Source
J. Appl. Probab., Volume 52, Number 1 (2015), 18-36.

Dates
First available in Project Euclid: 17 April 2015

Permanent link to this document
https://projecteuclid.org/euclid.jap/1429282604

Digital Object Identifier
doi:10.1239/jap/1429282604

Mathematical Reviews number (MathSciNet)
MR3336844

Zentralblatt MATH identifier
1315.60056

Subjects
Primary: 60G55: Point processes 26A33: Fractional derivatives and integrals
Secondary: 34A08: Fractional differential equations 60G22: Fractional processes, including fractional Brownian motion

Keywords
Dzhrbashyan-Caputo fractional derivative Poisson process stable process Mittag-Leffler function pure birth process

Citation

Garra, R.; Orsingher, E.; Polito, F. State-dependent fractional point processes. J. Appl. Probab. 52 (2015), no. 1, 18--36. doi:10.1239/jap/1429282604. https://projecteuclid.org/euclid.jap/1429282604


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