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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
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    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.
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