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State-Dependent Fractional Point Processes

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 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Applied Probability Cambridge University Press

State-Dependent Fractional Point Processes

Journal of Applied Probability , Volume 52 (1): 19 – Jan 30, 2018

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Publisher
Cambridge University Press
Copyright
© Applied Probability Trust 
ISSN
1475-6072
eISSN
0021-9002
DOI
10.1239/jap/1429282604
Publisher site
See Article on Publisher Site

Abstract

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.

Journal

Journal of Applied ProbabilityCambridge University Press

Published: Jan 30, 2018

Keywords: Dzhrbashyan-Caputo fractional derivative; Poisson process; stable process; Mittag-Leffler function; pure birth process; 60G55; 26A33; 34A08; 60G22

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