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Generalized Telegraph Process with Random Jumps

Part of: Special processes Markov processes

Published online by Cambridge University Press:  30 January 2018

Antonio Di Crescenzo ,
Antonella Iuliano ,
Barbara Martinucci  and
Shelemyahu Zacks
Antonio Di Crescenzo*
Affiliation:
Università di Salerno
Antonella Iuliano*
Affiliation:
Università di Salerno
Barbara Martinucci*
Affiliation:
Università di Salerno
Shelemyahu Zacks*
Affiliation:
Binghamton University
*
∗ Postal address: Dipartimento di Matematica, Università di Salerno, Via Giovanni Paolo II, n. 132, Fisciano (SA) 84084, Italy.
∗ Postal address: Dipartimento di Matematica, Università di Salerno, Via Giovanni Paolo II, n. 132, Fisciano (SA) 84084, Italy.
∗ Postal address: Dipartimento di Matematica, Università di Salerno, Via Giovanni Paolo II, n. 132, Fisciano (SA) 84084, Italy.
∗∗∗∗∗ Postal address: Department of Mathematical Sciences, Binghamton University, Binghamton, NY 13902-6000, USA. Email address: shelly@math.binghamton.edu
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Abstract

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We consider a generalized telegraph process which follows an alternating renewal process and is subject to random jumps. More specifically, consider a particle at the origin of the real line at time t=0. Then it goes along two alternating velocities with opposite directions, and performs a random jump toward the alternating direction at each velocity reversal. We develop the distribution of the location of the particle at an arbitrary fixed time t, and study this distribution under the assumption of exponentially distributed alternating random times. The cases of jumps having exponential distributions with constant rates and with linearly increasing rates are treated in detail.

Keywords

Jump-telegraph processalternating renewal processexponential random timerandom jump

MSC classification

Primary: 60K15: Markov renewal processes, semi-Markov processes
Secondary: 60J75: Jump processes
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 2 , June 2013 , pp. 450 - 463
Copyright
© Applied Probability Trust 

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