On the governing equations for Poisson and Skellam processes time-changed by inverse subordinators

Abstract

In the paper we present the governing equations for marginal distributions of Poisson and Skellam processes time-changed by inverse subordinators. The equations are given in terms of convolution-type derivatives.

Models of space-time random fields on the sphere

“…Remark 2. The proof of the above Proposition has been derived, by different approaches, in [15], [20], [6]. In particular, in [6], the authors consider the time-changed Poisson process N Lt (with L t being an inverse subordinator independet of the process N ) and show that its marginal distributions p x (t) = P(N Lt = x), x = 0, 1, .…”

Generalized fractional calculus and some models of generalized counting processes

“…In particular, in paper [21] a general class of time-changed Poisson processes N(H ψ (t)), t > 0, was introduced and studied, where N(t) is a Poisson process and H ψ (t) is an arbitrary subordinator with the Laplace exponent ψ, independent of N(t), and their distributional properties, hitting times and governing equations were presented (see, also [8]). In paper [5] Poisson processes time-changed by general inverse subordinators were studied, the governing equations for their marginal distributions were presented and some other properties were described. The Poisson process itself, being in a sense a core object concerning applicability to count data and simple tractability, however, as a reverse side of its simplicity, is a rather restrictive model.…”

Skellam Type Processes of Order k and Beyond

“…Similar observations are observed in the case of Danish fire insurance data (see [ 11 ]). Buchak and Sakhno, in [ 12 ], have also proposed the governing equations for time-fractional Skellam processes. Recently, [ 13 ] introduced time-changed Poisson process of order k , which is obtained by time changing the Poisson process of order k (see [ 14 ]) by general subordinators.…”