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On a jump-telegraph process driven by an alternating fractional Poisson process

Part of: Stochastic processes Special processes Markov processes

Published online by Cambridge University Press:  28 March 2018

Antonio Di Crescenzo  and
Alessandra Meoli
Antonio Di Crescenzo*
Affiliation:
Università di Salerno
Alessandra Meoli*
Affiliation:
Università di Salerno
*
* Postal address: Dipartimento di Matematica, Università di Salerno, Via Giovanni Paolo II, 132, 84084 Fisciano, SA, Italy.
* Postal address: Dipartimento di Matematica, Università di Salerno, Via Giovanni Paolo II, 132, 84084 Fisciano, SA, Italy.
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Abstract

The basic jump-telegraph process with exponentially distributed interarrival times deserves interest in various applied fields such as financial modelling and queueing theory. Aiming to propose a more general setting, we analyse such a stochastic process when the interarrival times separating consecutive velocity changes (and jumps) have generalized Mittag-Leffler distributions, and constitute the random times of a fractional alternating Poisson process. By means of renewal theory-based issues we obtain the forward and backward transition densities of the motion in series form, and prove their uniform convergence. Specific attention is then given to the case of jumps with constant size, for which we also obtain the mean of the process. Finally, we investigate the first-passage time of the process through a constant positive boundary, providing its formal distribution and suitable lower bounds.

Keywords

Finite velocityrandom motiongeneralized Mittag-Leffler distributionjump processfirst-passage time

MSC classification

Primary: 60K15: Markov renewal processes, semi-Markov processes
Secondary: 60G22: Fractional processes, including fractional Brownian motion 60J75: Jump processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 1 , March 2018 , pp. 94 - 111
Copyright
Copyright © Applied Probability Trust 2018 

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