Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

Probabilistic analysis of systems alternating for state-dependent dichotomous noise

Dipartimento di Matematica, Università degli Studi di Salerno, Via Giovanni Paolo II n. 132, 84084 Fisciano (SA), Italy

Aiming to construct a simple stochastic model able to describe systems alternating due to state-dependent dichotomous noise, we consider a generalized telegraph process whose sample-paths fluctuates around the zero state. Indeed, the latter process describes the motion of a particle on the real line, which is characterized by constant velocities and state-dependent intensities that vanish when the motion is toward the origin. This assumption allows to adopt an approach based on renewal theory to obtain formal expressions of the forward and backward transition densities of the process. The special case when certain random times of the motion possess gamma distribution leads to closed-form expressions of the transition densities, given in terms of the generalized Mittag-Leffler function. We also analyze a first-passage-time problem for the considered process in the presence of two constant boundaries.
  Article Metrics

Keywords telegraph process; random motion; intensity function; interarrival times; gamma distribution; generalized Mittag-Leffler function; first-passage-time problem, constant boundaries

Citation: Antonio Di Crescenzo, Fabio Travaglino. Probabilistic analysis of systems alternating for state-dependent dichotomous noise. Mathematical Biosciences and Engineering, 2019, 16(6): 6386-6405. doi: 10.3934/mbe.2019319


  • 1. I. Bena, Dichotomous Markov noise: exact results for out-of-equilibrium systems, Int. J. Modern Phys. B, 20 (2006), 2825–2888.
  • 2. P. Li, L. R. Nie, C. Z. Shu, et al., Effect of correlated dichotomous noises on stochastic resonance in a linear system, J. Stat. Phys., 146 (2012), 1184–1202.
  • 3. Y. Jin, Z. Ma and S. Xiao, Coherence and stochastic resonance in a periodic potential driven by multiplicative dichotomous and additive white noise, Chaos Solit. Fract., 103 (2017), 470–475.
  • 4. F. Müller-Hansen, F. Droste and B. Lindner, Statistics of a neuron model driven by asymmetric colored noise, Phys. Rev. E, 91 (2015), 022718.
  • 5. F. Droste and B. Lindner, Integrate-and-fire neurons driven by asymmetric dichotomous noise, Biol. Cybernet., 108 (2014), 825–843.
  • 6. R. Mankin and N. Lumi, Statistics of a leaky integrate-and-fire model of neurons driven by dichotomous noise, Phys. Rev. E, 93 (2016), 052143.
  • 7. S. Goldstein, On diffusion by discontinuous movements, and on the telegraph equation, Quart. J. Mech. Appl. Math., 4 (1951), 129–156.
  • 8. M. Kac, A stochastic model related to the telegrapher's equation, Rocky Mountain J. Math., 4 (1974), 497–509.
  • 9. A. D. Kolesnik and N. Ratanov, Telegraph Processes and Option Pricing, Springer Briefs in Statistics, Springer, Heidelberg, 2013.
  • 10. E. Orsingher, Probability law, flow function, maximum distribution of wave-governed random motions and their connections with Kirchoff's laws, Stoch. Process. Appl., 34 (1990), 49–66.
  • 11. 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), 11–25.
  • 12. O. López and N. Ratanov, On the asymmetric telegraph processes, J. Appl. Probab., 51 (2014), 569–589.
  • 13. D. Bshouty, A. Di Crescenzo, B. Martinucci, et al., Generalized telegraph process with random delays, J. Appl. Probab., 49 (2012), 850–865.
  • 14. A. Di Crescenzo and B. Martinucci, A damped telegraph random process with logistic stationary distribution, J. Appl. Probab., 47 (2010), 84–96.
  • 15. A. De Gregorio and C. Macci, Large deviations for a damped telegraph process, in Modern problems in insurance mathematics (eds. D. Silvestrov and A. Martin-Löf), EAA Ser., Springer, Cham, (2014), 275–289.
  • 16. A.D. Kolesnik, Probability distribution function for the Euclidean distance between two telegraph processes, Adv. Appl. Probab., 46 (2014), 1172–1193.
  • 17. A. D. Kolesnik, The explicit probability distribution of the sum of two telegraph processes, Stoch. Dyn., 15 (2015), 1550013.
  • 18. B. Martinucci and A. Meoli, Certain functionals of squared telegraph processes, Stoch. Dyn., (2019) Online Ready.
  • 19. A. Di Crescenzo, B. Martinucci and S. Zacks, Telegraph process with elastic boundary at the origin, Methodol. Comput. Appl. Probab., 20 (2018), 333–352.
  • 20. A. A. Pogorui, R. M. Rodr´ ıguez-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), 998–1007.
  • 21. N. Ratanov, Telegraph processes with random jumps and complete market models, Methodol. Comput. Appl. Probab., 17 (2015), 677–695.
  • 22. A. Di Crescenzo and S. Zacks, Probability law and flow function of Brownian motion driven by a generalized telegraph process, Methodol. Comput. Appl. Probab., 17 (2015), 761–780.
  • 23. F. Travaglino, A. Di Crescenzo, B. Martinucci, et al., A new model of Campi Flegrei inflation and deflation episodes based on Brownian motion driven by the telegraph process, Math. Geosci., 50 (2018), 961–975.
  • 24. J. Bierkens, P. Fearnhead and G. Roberts, The zig-zag process and super-efficient sampling for Bayesian analysis of big data, arXiv:1607.03188v2.
  • 25. S. M. Iacus, Statistical analysis of the inhomogeneous telegrapher's process, Statist. Probab. Lett., 55 (2001), 83–88.
  • 26. R. Garra and E. Orsingher, Random motions with space-varying velocities, in Modern problems of stochastic analysis and statistics (ed. V. Panov), Springer Proc. Math. Stat., 208, Springer, Cham, (2017), 25–39.
  • 27. G. D'Onofrio, P. Lansky and E. Pirozzi, On two diffusion neuronal models with multiplicative noise: The mean first-passage time properties, Chaos, 28 (2018), 043103.
  • 28. A. Di Crescenzo and B. Martinucci, Analysis of a stochastic neuronal model with excitatory inputs and state-dependent effects, Math. Biosci., 209 (2007), 547–563.
  • 29. A. Di Crescenzo, On random motions with velocities alternating at Erlang-distributed random times, Adv. Appl. Probab., 33 (2001), 690–701.
  • 30. W. Alharbi and S. Petrovskii, Critical domain problem for the reaction-telegraph equation model of population dynamics, Mathematics, 6 (2018), 59.
  • 31. L. Angelani and R. Garra, Probability distributions for the run-and-tumble models with variable speed and tumbling rate, Mod. Stoch. Theory Appl., 6 (2019), 3–12.
  • 32. R. Garcia, F. Moss, A. Nihongi, et al., Optimal foraging by zooplankton within patches: the case of Daphnia, Math. Biosci., 207 (2007), 165–188.
  • 33. A. Di Crescenzo and B. Martinucci, Random motion with gamma-distributed alternating velocities in biological modeling, in Computer Aided Systems Theory - Eurocast 2007, Lecture Notes in Computer Science (eds. R. Moreno-Diaz, F. Pichler and A. Quesada-Arencibia), Springer-Verlag, Berlin, (2007), 163–170.
  • 34. G. Le Caër, A new family of solvable Pearson-Dirichlet random walks, J. Stat. Phys., 144 (2011), 23–45.
  • 35. A. A. Pogorui and R.M. Rodríguez-Dagnino, Random motion with gamma steps in higher dimen-sions, Statist. Probab. Lett., 83 (2013), 638–1643.
  • 36. R. Gorenflo, A. A. Kilbas, F. Mainardi, et al., The two-parametric Mittag-Leffler function, in Mittag-Leffler functions, related topics and applications, Springer Monographs in Mathematics. Springer, Berlin, Heidelberg, (2014), 55–96.
  • 37. H. J. Haubold, A. M. Mathai and R. K. Saxena, Mittag-Leffler functions and their applications, J. Appl. Math., (2011), Art. ID 298628, 51 pp.
  • 38. M. Alipour, L. Beghin and D. Rostamy, Generalized fractional nonlinear birth processes, Methodol. Comput. Appl. Probab., 17 (2015), 525–540.
  • 39. A. Di Crescenzo, B. Martinucci and A. Meoli, A fractional counting process and its connection with the Poisson process, ALEA Lat. Am. J. Probab. Math. Stat., 13 (2016), 291–307.
  • 40. E. Orsingher and F. Polito, On a fractional linear birth-death process, Bernoulli, 17 (2011), 114–137.
  • 41. P. Lansky, L. Sacerdote and C. Zucca, Optimum signal in a diffusion leaky integrate-and-fire neuronal model, Math. Biosci., 207 (2007), 261–274.


Reader Comments

your name: *   your email: *  

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved