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A Generalized Telegraph Process with Velocity Driven by Random Trials

Part of: Special processes

Published online by Cambridge University Press:  04 January 2016

Irene Crimaldi ,
Antonio Di Crescenzo ,
Antonella Iuliano  and
Barbara Martinucci
Irene Crimaldi*
Affiliation:
IMT
Antonio Di Crescenzo*
Affiliation:
Università di Salerno
Antonella Iuliano*
Affiliation:
Istituto per le Applicazioni del Calcolo ‘Mauro Picone’
Barbara Martinucci*
Affiliation:
Università di Salerno
*
∗ Postal address: Institute for Advanced Studies, IMT, Piazza San Ponziano 6, I-55100 Lucca, Italy. Email address: irene.crimaldi@imtlucca.it
∗∗ Postal address: Dipartimento di Matematica, Università di Salerno, Via Ponte don Melillo, Fisciano (SA), I-84084, Italy.
∗∗∗∗ Postal address: Istituto per le Applicazioni del Calcolo ‘Mauro Picone’, CNR, Via Pietro Castellino 111, I-80131 Napoli, Italy. Email address: a.iuliano@na.iac.cnr.it
∗∗ Postal address: Dipartimento di Matematica, Università di Salerno, Via Ponte don Melillo, Fisciano (SA), I-84084, Italy.
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Abstract

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We consider a random trial-based telegraph process, which describes a motion on the real line with two constant velocities along opposite directions. At each epoch of the underlying counting process the new velocity is determined by the outcome of a random trial. Two schemes are taken into account: Bernoulli trials and classical Pólya urn trials. We investigate the probability law of the process and the mean of the velocity of the moving particle. We finally discuss two cases of interest: (i) the case of Bernoulli trials and intertimes having exponential distributions with linear rates (in which, interestingly, the process exhibits a logistic stationary density with nonzero mean), and (ii) the case of Pólya trials and intertimes having first gamma and then exponential distributions with constant rates.

Keywords

Telegraph processrandom intertimerandom velocityBernoulli schemePólya urn modellogistic stationary density

MSC classification

Primary: 60K15: Markov renewal processes, semi-Markov processes
Secondary: 60K37: Processes in random environments
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 45 , Issue 4 , December 2013 , pp. 1111 - 1136
Copyright
© Applied Probability Trust 

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