Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-12T19:48:26.361Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Asymmetric Telegraph Processes

Part of: Special processes Limit theorems Markov processes

Published online by Cambridge University Press:  19 February 2016

Oscar López  and
Nikita Ratanov
Oscar López*
Affiliation:
Universidad del Rosario
*
∗ Postal address: Universidad del Rosario, Cl. 12c, No. 4-69, Bogotá, Colombia.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study the one-dimensional random motion X = X(t), t ≥ 0, which takes two different velocities with two different alternating intensities. The closed-form formulae for the density functions of X and for the moments of any order, as well as the distributions of the first passage times, are obtained. The limit behaviour of the moments is analysed under nonstandard Kac's scaling.

Keywords

Asymmetric telegraph processfirst passage timeKac's asymptoticsmomentsmodified Bessel functionKummer function

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60J65: Brownian motion 60F05: Central limit and other weak theorems 60K99: None of the above, but in this section
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 2 , June 2014 , pp. 569 - 589
Copyright
© Applied Probability Trust 

References

Abramowitz, M. and Stegun, I. A. (eds) (1965). Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables. Dover, New York.Google Scholar
Beghin, L., Nieddu, L. and Orsingher, E. (2001). Probabilistic analysis of the telegrapher's process with drift by means of relativistic transformations. J. Appl. Math. Stoch. Anal. 14, 11–25.Google Scholar
Billingsley, P. (1986). Probability and Measure, 2nd edn. John Wiley, New York.Google Scholar
Bogachev, L. and Ratanov, N. (2011). Occupation time distributions for the telegraph process. Stoch. Process. Appl. 121, 1816–1844.Google Scholar
Bshouty, D., Di Crescenzo, A., Martinucci, B. and Zacks, S. (2012). Generalized telegraph process with random delays. J. Appl. Prob. 49, 850–865.Google Scholar
De Gregorio, A., Orsingher, E. and Sakhno, L. (2005). Motions with finite velocity analyzed with order statistics and differential equations. Theory Prob. Math. Statist. 71, 63–79.Google Scholar
Di Crescenzo, A. and Martinucci, B. (2013). On the generalized telegraph process with deterministic Jumps. Methodology Comput. Appl. Prob. 15, 215–235.Google Scholar
Foong, S. K. (1992). First-passage time, maximum displacement, and Kac's solution of the telegrapher equation. Phys. Rev. A. 46, 707–710.Google Scholar
Foong, S. K. and Kanno, S. (1994). Properties of the telegrapher's random process with or without a trap. Stoch. Process. Appl. 53, 147–173.Google Scholar
Hadeler, K. P. (1999). Reaction transport systems in biological modelling. In Mathematics Inspired by Biology (Lecture Notes in Math. Vol. 1714), eds Capasso, V. and Diekmann, O., Springer, Berlin, pp. 95–150.Google Scholar
Hillen, T. and Hadeler, K. P. (2005). Hyperbolic systems and transport equations in mathematical biology. In Analysis and Numerics for Conservation Laws, ed. Warnecke, G., Springer, Berlin, pp. 257–279.Google Scholar
Iacus, S. and Yoshida, N. (2009). Estimation for the discretely observed telegraph process. Theory Prob. Math. Statist. 78, 37–47.Google Scholar
Kac, M. (1974). A stochastic model related to the telegrapher's equation. Rocky Mountain J. Math. 4, 497–509.Google Scholar
Kolesnik, A. D. (2012). Moment analysis of the telegraph random process. Bull. Acad. Sci. Moldova, Ser. Math. 68, 90–107.Google Scholar
López, O. and Ratanov, N. (2012). Kac's rescaling for Jump-telegraph processes. Statist. Prob. Lett. 82, 1768–1776.Google Scholar
Okubo, A. and Levin, S. A. (2001). Diffusion and Ecological Problems: Modern Perspectives, 2nd edn. (Interdisciplinary Appl. Math. 14). Springer, New York.Google Scholar
Orsingher, E. (1990). Probability law, flow function, maximum distribution of wave-governed random motions and their connections with Kirchhoff's laws. Stoch. Process. Appl. 34, 49–66.CrossRefGoogle Scholar
Orsingher, E. (1995). Motions with reflecting and absorbing barriers driven by the telegraph equation. Random Operators Stoch. Equat. 3, 9–21.Google Scholar
Pinsky, M. A. (1991). Lectures on Random Evolution. World Scientific Publishing Co. River Edge, New Jersey.CrossRefGoogle Scholar
Ratanov, N. (1999). Telegraph evolutions in inhomogeneous media. Markov Process. Relat. Fields 5, 53–68.Google Scholar
Ratanov, N. (2007). A Jump telegraph model for option pricing. Quant. Finance 7, 575–583.Google Scholar
Stadje, W. and Zacks, S. (2004). Telegraph processes with random velocities. J. Appl. Prob. 41, 665–678.Google Scholar
Weiss, G. H. (2002). Some applications of persistent random walks and the telegrapher's equation. Physica A. 311, 381–410.Google Scholar
Zacks, S. (2004). Generalized integrated telegraph processes and the distribution of related stopping times. J. Appl. Prob. 41, 497–507.CrossRefGoogle Scholar