State-dependent fractional point processes. (English) Zbl 1315.60056

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

J. Appl. Probab. 52, No. 1, 18-36 (2015).

Summary: In this paper we analyse the fractional Poisson process where the state probabilities \(p_{k}^{\nu_{k}}(t)\), \(t \geq 0\), are governed by time-fractional equations of order \(0 < \nu_{k} \leq 1\) depending on the number \(k\) of events that have occurred up to time \(t\). We are able to obtain explicitly the Laplace transform of \(p_{k}^{\nu_{k}}(t)\) and various representations of state probabilities. We show that the Poisson process with intermediate waiting times depending on \(\nu_{k}\) differs from that constructed from the fractional state equations (in the case of \(\nu_{k} = \nu\) 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.

MSC:

60G55	Point processes (e.g., Poisson, Cox, Hawkes processes)
60G22	Fractional processes, including fractional Brownian motion
60G52	Stable stochastic processes
34A08	Fractional ordinary differential equations
26A33	Fractional derivatives and integrals

Keywords:

fractional point processes; Dzhrbashyan-Caputo fractional derivative; Poisson process; stable process; Mittag-Leffler function; fractional birth process

Software:

DLMF

References:

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