IDEAS home Printed from https://ideas.repec.org/a/gam/jrisks/v11y2023i7p126-d1190930.html
   My bibliography  Save this article

Cox-Based and Elliptical Telegraph Processes and Their Applications

Author

Listed:
  • Anatoliy Pogorui

    (Department of Mathematics, Zhytomyr Ivan Franko State University, Valyka Berdychivska St., 40, 10008 Zhytomyr, Ukraine)

  • Anatoly Swishchuk

    (Department of Mathematics & Statistics, University of Calgary, Calgary, AB T2N 1N4, Canada)

  • Ramón M. Rodríguez-Dagnino

    (School of Engineering and Sciences, Tecnologico de Monterrey, Av. Eugenio Garza Sada 2501 Sur, Monterrey 64849, Mexico)

  • Alexander Sarana

    (Department of Physics and Math Sciences, Zhytomyr Ivan Franko State University, Valyka Berdychivska St., 40, 10008 Zhytomyr, Ukraine)

Abstract

This paper studies two new models for a telegraph process: Cox-based and elliptical telegraph processes. The paper deals with the stochastic motion of a particle on a straight line and on an ellipse with random positive velocity and two opposite directions of motion, which is governed by a telegraph–Cox switching process. A relevant result of our analysis on the straight line is obtaining a linear Volterra integral equation of the first kind for the characteristic function of the probability density function (PDF) of the particle position at a given time. We also generalize Kac’s condition for the telegraph process to the case of a telegraph–Cox switching process. We show some examples of random velocity where the distribution of the coordinate of a particle is expressed explicitly. In addition, we present some novel results related to the switched movement evolution of a particle according to a telegraph–Cox process on an ellipse. Numerical examples and applications are presented for a telegraph–Cox-based process (option pricing formulas) and elliptical telegraph process.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jrisks:v:11:y:2023:i:7:p:126-:d:1190930
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-9091/11/7/126/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-9091/11/7/126/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. 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.
    2. Gitterman, M., 2005. "Classical harmonic oscillator with multiplicative noise," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 352(2), pages 309-334.
    Full references (including those not matched with items on IDEAS)

    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. 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. Nikita Ratanov, 2021. "Ornstein-Uhlenbeck Processes of Bounded Variation," Methodology and Computing in Applied Probability, Springer, vol. 23(3), pages 925-946, September.
    3. 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.
    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. Lin, Lifeng & Lin, Tianzhen & Zhang, Ruoqi & Wang, Huiqi, 2023. "Generalized stochastic resonance in a time-delay fractional oscillator with damping fluctuation and signal-modulated noise," Chaos, Solitons & Fractals, Elsevier, vol. 170(C).
    6. Chen, Xi & Luo, Maokang & Zhong, Yangfan & Zhang, Lu, 2022. "Collective dynamic behaviors of a general adjacent coupled chain in both unconfined and confined spaces," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 605(C).
    7. Tian, Yan & He, Guitian & Liu, Zhibin & Zhong, Linfeng & Yang, Xinping & Stanley, H. Eugene & Tu, Zhe, 2021. "The impact of memory effect on resonance behavior in a fractional oscillator with small time delay," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 563(C).
    8. 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.
    9. He, Lifang & Wu, Xia & Zhang, Gang, 2020. "Stochastic resonance in coupled fractional-order linear harmonic oscillators with damping fluctuation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 545(C).
    10. Du, Yuru & Meng, Lin & Lin, Lifeng & Wang, Huiqi, 2024. "Resonant behaviors of two coupled fluctuating-frequency oscillators with tempered Mittag-Leffler memory kernel," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 633(C).
    11. Berman, Michael & Cederbaum, Lorenz S., 2018. "Fractional driven-damped oscillator and its general closed form exact solution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 505(C), pages 744-762.
    12. Guo, Yong-Feng & Shen, Ya-Jun & Tan, Jian-Guo, 2015. "Effects of colored noise and external periodic force on the time derivative of information entropy for a damped harmonic oscillator," Applied Mathematics and Computation, Elsevier, vol. 252(C), pages 20-26.
    13. 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.
    14. Ratanov, Nikita, 2021. "On telegraph processes, their first passage times and running extrema," Statistics & Probability Letters, Elsevier, vol. 174(C).
    15. Cinque, Fabrizio, 2022. "A note on the conditional probabilities of the telegraph process," Statistics & Probability Letters, Elsevier, vol. 185(C).
    16. You, Pinlong & Lin, Lifeng & Wang, Huiqi, 2020. "Cooperative mechanism of generalized stochastic resonance in a time-delayed fractional oscillator with random fluctuations on both mass and damping," Chaos, Solitons & Fractals, Elsevier, vol. 135(C).
    17. 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.
    18. Lin, Lifeng & He, Minyue & Wang, Huiqi, 2022. "Collective resonant behaviors in two coupled fluctuating-mass oscillators with tempered Mittag-Leffler memory kernel," Chaos, Solitons & Fractals, Elsevier, vol. 154(C).
    19. Antonella Iuliano & Gabriella Verasani, 2024. "A Cyclic Random Motion in $$\mathbb {R}^3$$ R 3 Driven by Geometric Counting Processes," Methodology and Computing in Applied Probability, Springer, vol. 26(2), pages 1-23, June.

    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:gam:jrisks:v:11:y:2023:i:7:p:126-:d:1190930. 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: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    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.