Abstract
We consider a telegraph process with elastic boundary at the origin studied recently in the literature (see e.g. Di Crescenzo et al. (Methodol Comput Appl Probab 20:333–352 2018)). It is a particular random motion with finite velocity which starts at x ≥ 0, and its dynamics is determined by upward and downward switching rates λ and μ, with λ > μ, and an absorption probability (at the origin) α ∈ (0,1]. Our aim is to study the asymptotic behavior of the absorption time at the origin with respect to two different scalings: \(x\to \infty \) in the first case; \(\mu \to \infty \), with λ =β μ for some β > 1 and x > 0, in the second case. We prove several large and moderate deviation results. We also present numerical estimates of β based on an asymptotic Normality result for the case of the second scaling.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Crimaldi I, Di Crescenzo A, Iuliano A, Martinucci B (2013) A generalized telegraph process with velocity driven by random trials. Adv Appl Prob 45:1111–1136
Dembo A, Zeitouni O (1998) Large Deviations Techniques and Applications, vol 2. Springer, New York
Di Crescenzo A, Martinucci B (2010) A damped telegraph random process with logistic stationary distribution. J Appl Prob 47:84–96
Di Crescenzo A, Zacks S (2015) Probability law and flow function of Brownian motion driven by a generalized telegraph process. Methodol Comput Appl Probab 17:761–780
Di Crescenzo A, Martinucci B, Zacks S (2018) Telegraph process with elastic boundary at the origin. Methodol Comput Appl Probab 20:333–352
De Gregorio A, Orsingher E (2011) Flying randomly in \(\mathbb {R}^{d}\) with Dirichlet displacements. Stoch Proc Appl 122:676–713
Dominé M (1995) Moments of the first-passage time of a Wiener process with drift between two elastic barriers. J Appl Prob 32:1007–1013
Dominé M (1996) First passage time distribution of a Wiener process with drift concerning two elastic barriers. J Appl Prob 33:164–175
Foong SK (1992) First-passage time, maximum displacement, and Kac’s solution of the telegrapher equation. Phys Rev A 46:R707–R710
Garra R, Orsingher E (2014) Random flights governed by Klein-Gordon-type partial differential equations. Stoch Proc Appl 124:2171–2187
Giorno V, Nobile AG, Pirozzi E, Ricciardi LM (2006) On the construction of first-passage-time densities for diffusion processes. Sci Math Jpn 64:277–298
Goldstein S (1951) On diffusion by discontinuous movements, and on the telegraph equation. Quart J Mech Appl Math 4:129–156
Jacob E (2012) A Langevin process reflected at a partially elastic boundary: I. Stoch Proc Appl 122:191–216
Jacob E (2013) Langevin Process reflected on a partially elastic boundary II. séminaire de probabilités XLV:245–275: Lecture Notes in Mathm, 2078. Springer, Cham
Kac M (1974) A stochastic model related to the telegrapher’s equation. Rocky Mountain J Math 4:497–509
Macci C (2011) Large deviation results for wave governed random motions driven by semi-Markov processes. Comm Statist Simulation Comput 40:1342–1363
Macci C (2016) Large deviations for some non-standard telegraph processes. Statist Probab Lett 110:119–127
Orsingher E (1990) Probability law, flow function, maximum distribution of wave-governed random motions and their connections with Kirchoff’s laws. Stoch Proc Appl 34:49–66
Orsingher E (1995) Motions with reflecting and absorbing barriers driven by the telegraph equation. Random Oper Stochastic Equations 3:9–21
Ratanov NE (1997) Random walks in an inhomogeneous one-dimensional medium with reflecting and absorbing barriers. Theoret Math Phys 112:857–865
Ratanov N (2015) Telegraph processes with random jumps and complete market models. Methodol Comput Appl Probab 17:677–695
Stadje W, Zacks S (2004) Telegraph processes with random velocities. J Appl Prob 41:665–678
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The authors acknowledge the support of: GNAMPA and GNCS groups of INdAM (Istituto Nazionale di Alta Matematica); MIUR–PRIN 2017, Project ‘Stochastic Models for Complex Systems’ (no. 2017JFFHSH); MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata (CUP E83C18000100006).
Rights and permissions
About this article
Cite this article
Macci, C., Martinucci, B. & Pirozzi, E. Asymptotic Results for the Absorption Time of Telegraph Processes with Elastic Boundary at the Origin. Methodol Comput Appl Probab 23, 1077–1096 (2021). https://doi.org/10.1007/s11009-020-09804-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11009-020-09804-y