On the distribution of the maximum of the telegraph process
Authors:
F. Cinque and E. Orsingher
Journal:
Theor. Probability and Math. Statist. 102 (2020), 73-95
MSC (2020):
Primary 60K99
DOI:
https://doi.org/10.1090/tpms/1128
Published electronically:
March 29, 2021
Full-text PDF
Abstract |
References |
Similar Articles |
Additional Information
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.
References
- L. Beghin, L. Nieddu, and E. Orsingher, Probabilistic analysis of the telegrapher’s process with drift by means of relativistic transformations, J. Appl. Math. Stochastic Anal. 14 (2001), no. 1, 11–25. Special issue: Advances in applied stochastics. MR 1825098, DOI https://doi.org/10.1155/S104895330100003X
- Violet R. Cane, Diffusion models with relativity effects, Perspectives in probability and statistics (papers in honour of M. S. Bartlett on the occasion of his 65th birthday), Applied Probability Trust, Univ. Sheffield, Sheffield, 1975, pp. 263–273. MR 0408013, DOI https://doi.org/10.1017/s0021900200047707
- A. de Gregorio, E. Orsingher, and L. Sakhno, Motions with finite velocity analyzed with order statistics and differential equations, Teor. Ĭmovīr. Mat. Stat. 71 (2004), 57–71; English transl., Theory Probab. Math. Statist. 71 (2005), 63–79. MR 2144321, DOI https://doi.org/10.1090/S0094-9000-05-00648-4
- Antonio Di Crescenzo and Franco Pellerey, On prices’ evolutions based on geometric telegrapher’s process, Appl. Stoch. Models Bus. Ind. 18 (2002), no. 2, 171–184. MR 1907356, DOI https://doi.org/10.1002/asmb.456
- See Kit Foong, First-passage time, maximum displacement, and Kac’s solution of the telegrapher equation, Phys. Rev. A (3) 46 (1992), no. 2, R707–R710. MR 1175578, DOI https://doi.org/10.1103/PhysRevA.46.R707
- S. K. Foong and S. Kanno, Properties of the telegrapher’s random process with or without a trap, Stochastic Process. Appl. 53 (1994), no. 1, 147–173. MR 1290711, DOI https://doi.org/10.1016/0304-4149%2894%2990061-2
- Stanley Kaplan, Differential equations in which the Poisson process plays a role, Bull. Amer. Math. Soc. 70 (1964), 264–268. MR 158183, DOI https://doi.org/10.1090/S0002-9904-1964-11112-5
- A. A. Pogorui, R. M. Rodríguez-Dagnino, and T. Kolomiets, The first passage time and estimation of the number of level-crossings for a telegraph process, Ukrainian Math. J. 67 (2015), no. 7, 998–1007. Reprint of Ukraïn. Mat. Zh. 67 (2015), no. 7, 882–889. MR 3432862, DOI https://doi.org/10.1007/s11253-015-1132-y
- Enzo Orsingher, Probability law, flow function, maximum distribution of wave-governed random motions and their connections with Kirchoff’s laws, Stochastic Process. Appl. 34 (1990), no. 1, 49–66. MR 1039562, DOI https://doi.org/10.1016/0304-4149%2890%2990056-X
- W. Stadje and S. Zacks, Telegraph processes with random velocities, J. Appl. Probab. 41 (2004), no. 3, 665–678. MR 2074815, DOI https://doi.org/10.1239/jap/1091543417
- S. Zacks, Generalized integrated telegraph processes and the distribution of related stopping times, J. Appl. Probab. 41 (2004), no. 2, 497–507. MR 2052587, DOI https://doi.org/10.1239/jap/1082999081
References
- 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. MR 1825098
- V. Cane, Diffusion models with relativistic effects, J.Gani, ed., Perspectives in Probability and Statistics (Academic Press Appl. Probab. Trust, Sheffield, UK), (1975), 263–273. MR 0408013
- 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. MR 2144321
- A. Di Crescenzo and F. Pellerey, On prices’ evolutions based on geometric telegrapher’s process, Appl. Stoch. Models Bus. Ind. 18 (2002), 171-184. MR 1907356
- S.K. Foong, First passage time, maximum displacement and Kac’s solution of the telegrapher equation, Phys. Rev. A46 (1992), R707–R710. MR 1175578
- 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. MR 1290711
- S. Kaplan, Differential equations in which the Poisson process plays a role, Bull. Amer. Math. Soc. 70 (1964), 264–268. MR 158183
- 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. MR 3432862
- 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. MR 1039562
- W. Stadje and S. Zacks, Telegraph processes with random velocities, J. Appl. Probab. 41 (2004), 665–678. MR 2074815
- S. Zacks, Generalized integrated telegraph process and the distribution of related stopping times, J. Appl. Probab. 41 (2004), 497–507. MR 2052587
Similar Articles
Retrieve articles in Theory of Probability and Mathematical Statistics
with MSC (2020):
60K99
Retrieve articles in all journals
with MSC (2020):
60K99
Additional Information
F. Cinque
Affiliation:
Department of Statistical Sciences, Sapienza University of Rome, Italy
Email:
cinque.1700526@studenti.uniroma1.it
E. Orsingher
Affiliation:
Department of Statistical Sciences, Sapienza University of Rome, Italy
Email:
enzo.orsingher@uniroma1.it
Keywords:
Telegraph process,
induction principle,
reflection principle,
Bessel functions.
Received by editor(s):
January 22, 2020
Published electronically:
March 29, 2021
Article copyright:
© Copyright 2020
Taras Shevchenko National University of Kyiv