Journal Article

Cox-Based and Elliptical Telegraph Processes and Their Applications

Pogorui, Anatoliy, Swishchuk, Anatoly, Rodríguez-Dagnino, Ramón M., Sarana, Alexander

2023

Overview

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.

• Publisher: MDPI AG
• Subject: Cox process / Cox-based telegraph process / Distribution (Probability theory) / elliptical telegraph process / Fourier transforms / Kac’s condition / option pricing for Cox-based telegraph process / Probability distribution / Random variables / Securities prices / Stochastic processes / Telegraph / telegraph process / Tests, problems and exercises / Velocity