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Probability Law and Flow Function of Brownian Motion Driven by a Generalized Telegraph Process

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  • Antonio Di Crescenzo

    (Università di Salerno)

  • Shelemyahu Zacks

    (Binghamton University)

Abstract

We consider a standard Brownian motion whose drift alternates randomly between a positive and a negative value, according to a generalized telegraph process. We first investigate the distribution of the occupation time, i.e. the fraction of time when the motion moves with positive drift. This allows to obtain explicitly the probability law and the flow function of the random motion. We discuss three special cases when the times separating consecutive drift changes have (i) exponential distribution with constant rates, (ii) Erlang distribution, and (iii) exponential distribution with linear rates. In conclusion, in view of an application in environmental sciences we evaluate the density of a Wiener process with infinitesimal moments alternating at inverse Gaussian distributed random times.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:metcap:v:17:y:2015:i:3:d:10.1007_s11009-013-9392-1
    DOI: 10.1007/s11009-013-9392-1
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    References listed on IDEAS

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    Cited by:

    1. V. Pozdnyakov & L. M. Elbroch & C. Hu & T. Meyer & J. Yan, 2020. "On Estimation for Brownian Motion Governed by Telegraph Process with Multiple Off States," Methodology and Computing in Applied Probability, Springer, vol. 22(3), pages 1275-1291, September.
    2. Ratanov, Nikita, 2015. "Hypo-exponential distributions and compound Poisson processes with alternating parameters," Statistics & Probability Letters, Elsevier, vol. 107(C), pages 71-78.
    3. Vladimir Pozdnyakov & L. Mark Elbroch & Anthony Labarga & Thomas Meyer & Jun Yan, 2019. "Discretely Observed Brownian Motion Governed by Telegraph Process: Estimation," Methodology and Computing in Applied Probability, Springer, vol. 21(3), pages 907-920, September.
    4. 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).
    5. Macci, Claudio, 2016. "Large deviations for some non-standard telegraph processes," Statistics & Probability Letters, Elsevier, vol. 110(C), pages 119-127.
    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. 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.

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