Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-10T12:59:44.776Z Has data issue: false hasContentIssue false
  • English
  • Français

A Damped Telegraph Random Process with Logistic Stationary Distribution

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Antonio Di Crescenzo  and
Barbara Martinucci
Antonio Di Crescenzo*
Affiliation:
Università degli Studi di Salerno
Barbara Martinucci*
Affiliation:
Università degli Studi di Salerno
*
∗Postal address: Dipartimento di Matematica e Informatica, Università degli Studi di Salerno, Via Ponte don Melillo, I-84084 Fisciano (SA), Italy.
∗Postal address: Dipartimento di Matematica e Informatica, Università degli Studi di Salerno, Via Ponte don Melillo, I-84084 Fisciano (SA), Italy.
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 introduce a stochastic process that describes a finite-velocity damped motion on the real line. Differently from the telegraph process, the random times between consecutive velocity changes have exponential distribution with linearly increasing parameters. We obtain the probability law of the motion, which admits a logistic stationary limit in a special case. Various results on the distributions of the maximum of the process and of the first passage time through a constant boundary are also given.

Keywords

Alternating random processlogistic stationary distributiondistribution of the maximumfirst passage time

MSC classification

Secondary: 60K15: Markov renewal processes, semi-Markov processes 60K99: None of the above, but in this section
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 1 , March 2010 , pp. 84 - 96
Copyright
Copyright © Applied Probability Trust 2010 

Footnotes

This work has been partially supported by Regione Campania and MIUR (PRIN 2008).

References

[1] Abramowitz, M. and Stegun, I. A. (eds) (1992). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York.Google Scholar
[2] Di Crescenzo, A. (2001). On random motions with velocities alternating at Erlang-distributed random times. Adv. Appl. Prob. 33, 690–701.CrossRefGoogle Scholar
[3] Di Crescenzo, A. and Martinucci, B. (2007). Random motion with gamma-distributed alternating velocities in biological modeling. In Computer Aided Systems Theory - EUROCAST 2007 (Lecture Notes Comput. Sci. 4739), Springer, Berlin, pp. 163–170.Google Scholar
[4] Di Crescenzo, A. and Pellerey, F. (2002). On prices' evolutions based on geometric telegrapher's process. Appl. Stoch. Models Business Industry 18, 171–184.CrossRefGoogle Scholar
[5] Ferrante, P. (2009). Interloss time in M/M/1/1 loss system. J. Appl. Math. Stoch. Analysis, 14 pp.Google Scholar
[6] Foong, S. K. (1992). First-passage time, maximum displacement, and Kac's solution of the telegrapher equation. Phys. Rev. A 46, R707–R710.Google Scholar
[7] Gupta, R. D. and Kundu, D. (2007). Generalized exponential distribution: existing results and some recent developments. J. Statist. Planning Infer. 137, 3537–3547.Google Scholar
[8] Iacus, S. M. (2001). Statistical analysis of the inhomogeneous telegrapher's process. Statist. Prob. Lett. 55, 83–88.CrossRefGoogle Scholar
[9] Kolesnik, A. (1998). The equations of Markovian random evolution on the line. J. Appl. Prob. 35, 27–35.Google Scholar
[10] Lachal, A. (2006). Cyclic random motions in {R}{d}-space with n directions. ESAIM Prob. Statist. 10, 277–316.Google Scholar
[11] Li, M. (2009). A damped diffusion framework for financial modeling and closed-form maximum likelihood estimation. J. Econom. Dynamics Control 34, 132–157.Google Scholar
[12] Lin, G. D. and Hu, C.-Y. (2008). On characterizations of the logistic distribution. J. Statist. Planning Infer. 138, 1147–1156.Google Scholar
[13] Masoliver, J. (1993). Second-order dichotomous processes: damped free motion, critical behavior, and anomalous superdiffusion. Phys. Rev. E 48, 121–135.CrossRefGoogle ScholarPubMed
[14] Oluyede, B. O. (2004). Inequalities and comparisons of the Cauchy, Gauss, and logistic distributions. Appl. Math. Lett. 17, 77–82.CrossRefGoogle Scholar
[15] Orsingher, E. (1985). Hyperbolic equations arising in random models. Stoch. Process. Appl. 21, 93–106.Google Scholar
[16] Orsingher, E. (1990). Probability law, flow function, maximum distribution of wave-governed random motions and their connections with Kirchoff's laws. Stoch. Process. Appl. 34, 49–66.CrossRefGoogle Scholar
[17] Orsingher, E. (1995). Motions with reflecting and absorbing barriers driven by the telegraph equation. Random Operators Stoch. Equat. 3, 9–21.Google Scholar
[18] Orsingher, E. and Bassan, B. (1992). On a 2n-valued telegraph signal and the related integrated process. Stoch. Stoch. Reports 38, 159–173.CrossRefGoogle Scholar
[19] Pogorui, A. A. and Rodríguez-Dagnino, R. M. (2005). One-dimensional semi-Markov evolutions with general Erlang sojourn times. Random Operators Stoch. Equat. 13, 399–405.Google Scholar
[20] Pyke, R. (1965). Spacings. (With discussion.) J. R. Statist. Soc. B 27, 395–449.Google Scholar
[21] Ratanov, N. and Melnikov, A. (2008). On financial markets based on telegraph processes. Stochastics 80, 247–268.Google Scholar
[22] Stadje, W. and Zacks, S. (2004). Telegraph processes with random velocities. J. Appl. Prob. 41, 665–678.Google Scholar
[23] Van Lieshout, M. N. M. (2006). Maximum likelihood estimation for random sequential adsorption. Adv. Appl. Prob. 38, 889–898.Google Scholar
[24] Zacks, S. (2004). Generalized integrated telegraph processes and the distribution of related stopping times. J. Appl. Prob. 41, 497–507.Google Scholar