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Asymptotic Results for the Absorption Time of Telegraph Processes with Elastic Boundary at the Origin

Author

Listed:
  • Claudio Macci

    (Università di Roma Tor Vergata)

  • Barbara Martinucci

    (Università degli Studi di Salerno)

  • Enrica Pirozzi

    (Università di Napoli Federico II)

Abstract

We consider a telegraph process with elastic boundary at the origin studied recently in the literature (see e.g. Di Crescenzo et al. (Methodol Comput Appl Probab 20:333–352 2018)). It is a particular random motion with finite velocity which starts at x ≥ 0, and its dynamics is determined by upward and downward switching rates λ and μ, with λ > μ, and an absorption probability (at the origin) α ∈ (0,1]. Our aim is to study the asymptotic behavior of the absorption time at the origin with respect to two different scalings: x → ∞ $x\to \infty $ in the first case; μ → ∞ $\mu \to \infty $ , with λ =β μ for some β > 1 and x > 0, in the second case. We prove several large and moderate deviation results. We also present numerical estimates of β based on an asymptotic Normality result for the case of the second scaling.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:metcap:v:23:y:2021:i:3:d:10.1007_s11009-020-09804-y
    DOI: 10.1007/s11009-020-09804-y
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    References listed on IDEAS

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    1. 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.
    2. Macci, Claudio, 2016. "Large deviations for some non-standard telegraph processes," Statistics & Probability Letters, Elsevier, vol. 110(C), pages 119-127.
    3. 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.
    4. 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.
    5. Nikita Ratanov, 2015. "Telegraph Processes with Random Jumps and Complete Market Models," Methodology and Computing in Applied Probability, Springer, vol. 17(3), pages 677-695, September.
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    Cited by:

    1. 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).

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