IDEAS home Printed from https://ideas.repec.org/a/eee/stapro/v185y2022ics0167715222000384.html
   My bibliography  Save this article

A note on the conditional probabilities of the telegraph process

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167715222000384
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spl.2022.109431?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    2. 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.
    3. Alessandro Gregorio & Stefano Iacus, 2008. "Parametric estimation for the standard and geometric telegraph process observed at discrete times," Statistical Inference for Stochastic Processes, Springer, vol. 11(3), pages 249-263, October.
    4. 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.
    5. Nikita Ratanov, 2007. "Jump Telegraph Processes and Financial Markets with Memory," International Journal of Stochastic Analysis, Hindawi, vol. 2007, pages 1-19, October.
    6. 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.
    7. 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.
    8. L. Beghin & L. Nieddu & E. Orsingher, 2001. "Probabilistic analysis of the telegrapher's process with drift by means of relativistic transformations," International Journal of Stochastic Analysis, Hindawi, vol. 14, pages 1-15, January.
    9. 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.
    10. 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.
    11. Antonio Di Crescenzo & Franco Pellerey, 2002. "On prices' evolutions based on geometric telegrapher's process," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 18(2), pages 171-184, April.
    12. Ratanov, Nikita, 2021. "On telegraph processes, their first passage times and running extrema," Statistics & Probability Letters, Elsevier, vol. 174(C).
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. De Gregorio, Alessandro & Iafrate, Francesco, 2021. "Telegraph random evolutions on a circle," Stochastic Processes and their Applications, Elsevier, vol. 141(C), pages 79-108.
    3. Cinque, Fabrizio & Orsingher, Enzo, 2023. "Random motions in R3 with orthogonal directions," Stochastic Processes and their Applications, Elsevier, vol. 161(C), pages 173-200.
    4. Nikita Ratanov, 2020. "First Crossing Times of Telegraph Processes with Jumps," Methodology and Computing in Applied Probability, Springer, vol. 22(1), pages 349-370, March.
    5. Iuliano, Antonella & Macci, Claudio, 2023. "Asymptotic results for the absorption time of telegraph processes with a non-standard barrier at the origin," Statistics & Probability Letters, Elsevier, vol. 196(C).
    6. 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.
    7. Ratanov, Nikita, 2021. "On telegraph processes, their first passage times and running extrema," Statistics & Probability Letters, Elsevier, vol. 174(C).
    8. Nikita Ratanov, 2021. "Ornstein-Uhlenbeck Processes of Bounded Variation," Methodology and Computing in Applied Probability, Springer, vol. 23(3), pages 925-946, September.
    9. Iacus, Stefano Maria, 2001. "Statistical analysis of the inhomogeneous telegrapher's process," Statistics & Probability Letters, Elsevier, vol. 55(1), pages 83-88, November.
    10. Antonio Di Crescenzo & Shelemyahu Zacks, 2015. "Probability Law and Flow Function of Brownian Motion Driven by a Generalized Telegraph Process," Methodology and Computing in Applied Probability, Springer, vol. 17(3), pages 761-780, September.
    11. Claudio Macci & Barbara Martinucci & Enrica Pirozzi, 2021. "Asymptotic Results for the Absorption Time of Telegraph Processes with Elastic Boundary at the Origin," Methodology and Computing in Applied Probability, Springer, vol. 23(3), pages 1077-1096, September.
    12. De Gregorio, Alessandro & Macci, Claudio, 2012. "Large deviation principles for telegraph processes," Statistics & Probability Letters, Elsevier, vol. 82(11), pages 1874-1882.
    13. 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.
    14. Mazza, Christian & Rulliere, Didier, 2004. "A link between wave governed random motions and ruin processes," Insurance: Mathematics and Economics, Elsevier, vol. 35(2), pages 205-222, October.
    15. 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.
    16. Bogachev, Leonid & Ratanov, Nikita, 2011. "Occupation time distributions for the telegraph process," Stochastic Processes and their Applications, Elsevier, vol. 121(8), pages 1816-1844, August.
    17. Nikita Ratanov & Mikhail Turov, 2023. "On Local Time for Telegraph Processes," Mathematics, MDPI, vol. 11(4), pages 1-12, February.
    18. Soham Biswas & Francois Leyvraz, 2021. "Ballistic annihilation in one dimension: a critical review," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 94(12), pages 1-10, December.
    19. Anatoliy Pogorui & Anatoly Swishchuk & Ramón M. Rodríguez-Dagnino & Alexander Sarana, 2023. "Cox-Based and Elliptical Telegraph Processes and Their Applications," Risks, MDPI, vol. 11(7), pages 1-15, July.
    20. Thomas M. Michelitsch & Federico Polito & Alejandro P. Riascos, 2023. "Semi-Markovian Discrete-Time Telegraph Process with Generalized Sibuya Waiting Times," Mathematics, MDPI, vol. 11(2), pages 1-20, January.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:stapro:v:185:y:2022:i:c:s0167715222000384. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.