Theory of Probability and Mathematical Statistics
On the distribution of the maximum of the telegraph process
F. Cinque, E. Orsingher
Link
Abstract: In this paper we present the distribution of the maximum of the telegraph process in the cases where the initial velocity is positive or negative with an even and an odd number of velocity reversals. For the telegraph process with positive initial velocity a reflection principle is proved to be valid while in the case of an initial leftward displacement the conditional distributions are perturbed by a positive probability of never visiting the half positive axis. Various relationships are established among the mentioned four classes of conditional distributions of the maximum. The unconditional distributions of the maximum of the telegraph process are obtained for positive and negative initial steps as well as their limiting behaviour. Furthermore the cumulative distributions and the general moments of the conditional maximum are presented.
Keywords: Telegraph process, induction principle, reflection principle, Bessel functions
Bibliography: 1. L. Beghin, L. Nieddu and E. Orsingher, Probabilistic analysis of the telegrapher's process with drift by means of relativistic transformations, Journal of Applied Mathematics and Stochastic Analysis 14 (2001), 11-25.
2. V. Cane, Diffusion models with relativistic effects, J.Gani, ed., Perspectives in Probability and Statistics (Academic Press Appl. Probab. Trust, Shefield, UK), (1975), 263-273.
3. A. De Gregorio, E. Orsingher and L. Sakhno, Motions with finite velocity analyzed with order statistics and differential equations, Theory Probab. Math. Statist. 71 (2004), 63-79.
4. A. Di Crescenzo and F. Pellerey, On prices' evolutions based on geometric telegrapher's process, Appl. Stoch. Models Bus. Ind. 18 (2002), 171-184.
5. S. K. Foong, First passage time, maximum displacement and Kac's solution of the telegrapher equation, Phys. Rev. A46 (1992), R707-R710.
6. S. K. Foong and S. Kanno, Properties of the telegrapher's random process with or without a trap, Stochastic Process. Appl. 53 (1994), 147-173.
7. S. Kaplan, Differential equations in which the Poisson process plays a role, Bull. Amer. Math. Soc. 70 (1964), 264-268.
8. A. A. Pogorui, R.M. Rodriguez-Dagnino and T. Kolomiets, The first passage time and estimation of the number of level-crossings for a telegraph process, Ukr. Math. J. 67 (2015), 998-1007.
9. E. Orsingher, Probability law,
flow function, maximum distribution of wave-governed random motions and their connections with Kirchoff's laws, Stochastic Process. Appl. 34 (1990), 49-66.
10. W. Stadje and S. Zacks, Telegraph processes with random velocities, J. Appl. Probab. 41 (2004), 665-678.
11. S. Zacks, Generalized integrated telegraph process and the distribution of related stopping times, J. Appl. Probab. 41 (2004), 497-507.
