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A note on the conditional probabilities of the telegraph process

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  • Cinque, Fabrizio

Abstract

We consider the telegraph process with two velocities, a1>a2∈R, and two rates of reversal, λ1,λ2>0. We study some of its features with respect to the conditional probability measure where both the initial speed and the number of changes of direction are known. We exhibit a new proof by induction of the (conditional) probability law and a detailed study of the distribution of the motion at time t>0 conditioned on its position at a previous time 00, its maximum and its minimum up to that moment.

Suggested Citation

  • Cinque, Fabrizio, 2022. "A note on the conditional probabilities of the telegraph process," Statistics & Probability Letters, Elsevier, vol. 185(C).
    • Handle: RePEc:eee:stapro:v:185:y:2022:i:c:s0167715222000384
    DOI: 10.1016/j.spl.2022.109431
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    References listed on IDEAS

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    1. Foong, S. K. & Kanno, S., 1994. "Properties of the telegrapher's random process with or without a trap," Stochastic Processes and their Applications, Elsevier, vol. 53(1), pages 147-173, September.
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    6. Cinque, Fabrizio & Orsingher, Enzo, 2021. "On the exact distributions of the maximum of the asymmetric telegraph process," Stochastic Processes and their Applications, Elsevier, vol. 142(C), pages 601-633.
    7. Antonio Crescenzo & Barbara Martinucci & Paola Paraggio & Shelemyahu Zacks, 2021. "Some Results on the Telegraph Process Confined by Two Non-Standard Boundaries," Methodology and Computing in Applied Probability, Springer, vol. 23(3), pages 837-858, September.
    8. Antonio Di Crescenzo & Barbara Martinucci & Shelemyahu Zacks, 2018. "Telegraph Process with Elastic Boundary at the Origin," Methodology and Computing in Applied Probability, Springer, vol. 20(1), pages 333-352, March.
    9. Orsingher, Enzo, 1990. "Probability law, flow function, maximum distribution of wave-governed random motions and their connections with Kirchoff's laws," Stochastic Processes and their Applications, Elsevier, vol. 34(1), pages 49-66, February.
    10. Kolesnik, Alexander D. & Turbin, Anatoly F., 1998. "The equation of symmetric Markovian random evolution in a plane," Stochastic Processes and their Applications, Elsevier, vol. 75(1), pages 67-87, June.
    11. Antonio Di Crescenzo & Barbara Martinucci, 2013. "On the Generalized Telegraph Process with Deterministic Jumps," Methodology and Computing in Applied Probability, Springer, vol. 15(1), pages 215-235, March.
    12. Ratanov, Nikita, 2021. "On telegraph processes, their first passage times and running extrema," Statistics & Probability Letters, Elsevier, vol. 174(C).
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    Cited by:

    1. Nikita Ratanov & Mikhail Turov, 2023. "On Local Time for Telegraph Processes," Mathematics, MDPI, vol. 11(4), pages 1-12, February.
    2. Cinque, Fabrizio & Orsingher, Enzo, 2023. "Random motions in R3 with orthogonal directions," Stochastic Processes and their Applications, Elsevier, vol. 161(C), pages 173-200.

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