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

Abstract

We consider some fractional extensions of the recursive differential equation governing the Poisson process, i.e. d/dtpk(t) = -λ(pk(t) - pk - 1(t)), k ≥ 0, t > 0 by introducing fractional time-derivatives of order v, 2v, … , nv. We show that the so-called “Generalized Mittag-Leffler functions” Ekα,β (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 → 1. On the other hand, near the origin the behavior of the law of the interarrival times drastically changes for the parameter v varying in (0, 1]. For integer values of v, these models can be viewed as a higher-order Poisson processes, connected with the standard case by simple and explict relationships. © 2010 Applied Probability Trust.