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First Crossing Times of Telegraph Processes with Jumps

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  • Nikita Ratanov

    (Universidad del Rosario)

Abstract

The paper presents exact formulae related to the distribution of the first passage time τx of the jump-telegraph process. In particular, the Laplace transform of τx is analysed, when a jump component is in the opposite direction to the crossing level x > 0. The case of double exponential jumps is also studied in detail.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:metcap:v:22:y:2020:i:1:d:10.1007_s11009-019-09709-5
    DOI: 10.1007/s11009-019-09709-5
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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.
    2. Nikita Ratanov, 2007. "A jump telegraph model for option pricing," Quantitative Finance, Taylor & Francis Journals, vol. 7(5), pages 575-583.
    3. 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.
    4. Nikita Ratanov, 2008. "Option Pricing Model Based on a Markov-modulated Diffusion with Jumps," Papers 0812.0761, arXiv.org.
    5. Fontbona, Joaquin & Guérin, Hélène & Malrieu, Florent, 2016. "Long time behavior of telegraph processes under convex potentials," Stochastic Processes and their Applications, Elsevier, vol. 126(10), pages 3077-3101.
    6. Ratanov, Nikita, 2014. "On piecewise linear processes," Statistics & Probability Letters, Elsevier, vol. 90(C), pages 60-67.
    7. López, Oscar & Ratanov, Nikita, 2012. "Kac’s rescaling for jump-telegraph processes," Statistics & Probability Letters, Elsevier, vol. 82(10), pages 1768-1776.
    8. S. G. Kou, 2002. "A Jump-Diffusion Model for Option Pricing," Management Science, INFORMS, vol. 48(8), pages 1086-1101, August.
    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. 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.
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    Cited by:

    1. Nikita Ratanov, 2021. "Ornstein-Uhlenbeck Processes of Bounded Variation," Methodology and Computing in Applied Probability, Springer, vol. 23(3), pages 925-946, September.

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