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On the Generalized Telegraph Process with Deterministic Jumps

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  • Antonio Di Crescenzo

    (Università di Salerno)

  • Barbara Martinucci

    (Università di Salerno)

Abstract

We consider a semi-Markovian generalization of the integrated telegraph process subject to jumps. It describes a motion on the real line characterized by two alternating velocities with opposite directions, where a jump along the alternating direction occurs at each velocity reversal. We obtain the formal expressions of the forward and backward transition densities of the motion. We express them as series in the case of Erlang-distributed random times separating consecutive jumps. Furthermore, a closed form of the transition density is given for exponentially distributed times, with constant jumps and random initial velocity. In this case we also provide mean and variance of the process, and study the limiting behaviour of the probability law, which leads to a mixture of three Gaussian densities.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:metcap:v:15:y:2013:i:1:d:10.1007_s11009-011-9235-x
    DOI: 10.1007/s11009-011-9235-x
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    References listed on IDEAS

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    1. Nikita Ratanov, 2007. "A jump telegraph model for option pricing," Quantitative Finance, Taylor & Francis Journals, vol. 7(5), pages 575-583.
    2. 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.
    3. 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.
    4. D. Perry & W. Stadje & S. Zacks, 2005. "A Two-Sided First-Exit Problem for a Compound Poisson Process with a Random Upper Boundary," Methodology and Computing in Applied Probability, Springer, vol. 7(1), pages 51-62, March.
    5. Nikita Ratanov & Alexander Melnikov, 2007. "On Financial Markets Based on Telegraph Processes," Papers 0712.3428, arXiv.org.
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    Cited by:

    1. De Gregorio, Alessandro & Iafrate, Francesco, 2021. "Telegraph random evolutions on a circle," Stochastic Processes and their Applications, Elsevier, vol. 141(C), pages 79-108.
    2. 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.
    3. 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.
    4. 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.
    5. Cinque, Fabrizio, 2022. "A note on the conditional probabilities of the telegraph process," Statistics & Probability Letters, Elsevier, vol. 185(C).
    6. 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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