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 Electronic Journal of Probability > Vol. 15(2010) > Paper 22 open journal systems 


Poisson-type processes governed by fractional and higher-order recursive differential equations

Luisa Beghin, Sapienza University of Rome
Enzo Orsingher, Sapienza University of Rome


Abstract
We consider some fractional extensions of the recursive differential equation governing the Poisson process, i.e.
(d/(dt))pk(t)=-λ(pk(t)-pk-1(t)), k≥0,t>0
by introducing fractional time-derivatives of order ν, 2ν,...,nν. We show that the so-called "Generalized Mittag-Leffler functions" Eα,βk(x), x∈R (introduced by Prabhakar [24] )arise as solutions of these equations. The corresponding processes are proved to be renewal, with density of the intearrival times (represented by Mittag-Leffler functions) possessing power, instead of exponential, decay, for t→∞. On the other hand, near the origin the behavior of the law of the interarrival times drastically changes for the parameter ν varying in (0,1]. For integer values of ν, these models can be viewed as a higher-order Poisson processes, connected with the standard case by simple and explict relationships.


Full text: PDF

Pages: 684-709

Published on: May 20, 2010
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Electronic Journal of Probability. ISSN: 1083-6489