Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-27T12:28:15.841Z Has data issue: false hasContentIssue false
  • English
  • Français

On a jump-telegraph process driven by an alternating fractional Poisson process

Part of: Stochastic processes Special processes Markov processes

Published online by Cambridge University Press:  28 March 2018

Antonio Di Crescenzo  and
Alessandra Meoli
Antonio Di Crescenzo*
Affiliation:
Università di Salerno
Alessandra Meoli*
Affiliation:
Università di Salerno
*
* Postal address: Dipartimento di Matematica, Università di Salerno, Via Giovanni Paolo II, 132, 84084 Fisciano, SA, Italy.
* Postal address: Dipartimento di Matematica, Università di Salerno, Via Giovanni Paolo II, 132, 84084 Fisciano, SA, Italy.
Get access Rights & Permissions [Opens in a new window]

Abstract

The basic jump-telegraph process with exponentially distributed interarrival times deserves interest in various applied fields such as financial modelling and queueing theory. Aiming to propose a more general setting, we analyse such a stochastic process when the interarrival times separating consecutive velocity changes (and jumps) have generalized Mittag-Leffler distributions, and constitute the random times of a fractional alternating Poisson process. By means of renewal theory-based issues we obtain the forward and backward transition densities of the motion in series form, and prove their uniform convergence. Specific attention is then given to the case of jumps with constant size, for which we also obtain the mean of the process. Finally, we investigate the first-passage time of the process through a constant positive boundary, providing its formal distribution and suitable lower bounds.

Keywords

Finite velocityrandom motiongeneralized Mittag-Leffler distributionjump processfirst-passage time

MSC classification

Primary: 60K15: Markov renewal processes, semi-Markov processes
Secondary: 60G22: Fractional processes, including fractional Brownian motion 60J75: Jump processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 1 , March 2018 , pp. 94 - 111
Copyright
Copyright © Applied Probability Trust 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Beghin, L. and Orsingher, E. (2003). The telegraph process stopped at stable-distributed times and its connection with the fractional telegraph equation. Fract. Calculus Appl. Anal. 6, 187–204. Google Scholar
[2]Beghin, L. and Orsingher, E. (2009). Fractional Poisson processes and related planar random motions. Electron. J. Prob. 14, 1790–1827. CrossRefGoogle Scholar
[3]Beghin, L. and Orsingher, E. (2010). Poisson-type processes governed by fractional and higher-order recursive differential equations. Electron. J. Prob. 15, 684–709. Google Scholar
[4]Christoph, G. and Schreiber, K. (2001). Positive Linnik and discrete Linnik distributions. In Asymptotic Methods in Probability and Statistics with Applications, Birkhäuser, Boston, MA, pp. 3–17. CrossRefGoogle Scholar
[5]De Gregorio, A. and Iacus, S. M. (2008). Parametric estimation for the standard and geometric telegraph process observed at discrete times. Statist. Inference Stoch. Process. 11, 249–263. Google Scholar
[6]De Gregorio, A. and Iacus, S. M. (2011). Least-squares change-point estimation for the telegraph process observed at discrete times. Statistics 45, 349–359. Google Scholar
[7]Di Crescenzo, A. (2001). On random motions with velocities alternating at Erlang-distributed random times. Adv. Appl. Prob. 33, 690–701. Google Scholar
[8]Di Crescenzo, A. and Martinucci, B. (2010). A damped telegraph random process with logistic stationary distribution. J. Appl. Prob. 47, 84–96. Google Scholar
[9]Di Crescenzo, A. and Martinucci, B. (2013). On the generalized telegraph process with deterministic jumps. Methodol. Comput. Appl. Prob. 15, 215–235. Google Scholar
[10]Di Crescenzo, A. and Meoli, A. (2016). On a fractional alternating Poisson process. AIMS Math. 1, 212–224. Google Scholar
[11]Di Crescenzo, A., Martinucci, B. and Meoli, A. (2016). A fractional counting process and its connection with the Poisson process. ALEA Latin Amer. J. Prob. Math. Statist. 13, 291–307. CrossRefGoogle Scholar
[12]Di Crescenzo, A., Iuliano, A., Martinucci, B. and Zacks, S. (2013). Generalized telegraph process with random jumps. J. Appl. Prob. 50, 450–463. Google Scholar
[13]Ferraro, S., Manzini, M., Masoero, A. and Scalas, E. (2009). A random telegraph signal of Mittag-Leffler type. Physica A 388, 3991–3999. Google Scholar
[14]Garcia, R.et al. (2007). Optimal foraging by zooplankton within patches: the case of Daphnia. Math. Biosci. 207, 165–188. Google Scholar
[15]Goldstein, S. (1951). On diffusion by discontinuous movements, and on the telegraph equation. Quart. J. Mech. Appl. Math. 4, 129–156. Google Scholar
[16]Gorenflo, R., Kilbas, A. A., Mainardi, F. and Rogosin, S. V. (2014). Mittag-Leffler Functions, Related Topics and Applications. Springer, Heidelberg. CrossRefGoogle Scholar
[17]Iacus, S. M. and Yoshida, N. (2009). Estimation for the discretely observed telegraph process. Theory Prob. Math. Statist. 78, 37–47. CrossRefGoogle Scholar
[18]Jose, K. K., Uma, P., Lekshmi, V. S. and Haubold, H. J. (2010). Generalized Mittag-Leffler distributions and processes for applications in astrophysics and time series modeling. In Proceedings of the Third UN/ESA/NASA Workshop on the International Heliophysical Year 2007 and Basic Space Science, Springer, Heidelberg, pp. 79–92. Google Scholar
[19]Kac, M. (1974). A stochastic model related to the telegrapher's equation. Rocky Mountain J. Math. 4, 497–509. CrossRefGoogle Scholar
[20]Kilbas, A. A., Saigo, M. and Saxena, R. K. (2004). Generalized Mittag-Leffler function and generalized fractional calculus operators. Integral Transforms Special Functions 15, 31–49. Google Scholar
[21]Kolesnik, A. D. and Ratanov, N. (2013). Telegraph Processes and Option Pricing. Springer, Heidelberg. Google Scholar
[22]López, O. and Ratanov, N. (2012). Kac's rescaling for jump-telegraph processes. Statist. Prob. Lett. 82, 1768–1776. Google Scholar
[23]López, O. and Ratanov, N. (2014). On the asymmetric telegraph processes. J. Appl. Prob. 51, 569–589. CrossRefGoogle Scholar
[24]Masoliver, J. (2016). Fractional telegrapher's equation from fractional persistent random walks. Phys. Rev. E 93, 052107. Google Scholar
[25]Mathai, A. M. and Haubold, H. J. (2008). Special Functions for Applied Scientists. Springer, New York. Google Scholar
[26]Orsingher, E. and Beghin, L. (2004). Time-fractional telegraph equations and telegraph processes with Brownian time. Prob. Theory Relat. Fields 128, 141–160. Google Scholar
[27]Orsingher, E. and Polito, F. (2010). Fractional pure birth processes. Bernoulli 16, 858–881. Google Scholar
[28]Orsingher, E. and Polito, F. (2011). On a fractional linear birth-death process. Bernoulli 17, 114–137. Google Scholar
[29]Orsingher, E. and Zhao, X. (2003). The space-fractional telegraph equation and the related fractional telegraph process. Chinese Ann. Math. B 24, 45–56. Google Scholar
[30]Orsingher, E., Polito, F. and Sakhno, L. (2010). Fractional non-linear, linear and sublinear death processes. J. Statist. Phys. 141, 68–93. Google Scholar
[31]Othmer, H. G., Dunbar, S. R. and Alt, W. (1988). Models of dispersal in biological systems. J. Math. Biol. 26, 263–298. CrossRefGoogle ScholarPubMed
[32]Pakes, A. G. (1995). Characterization of discrete laws via mixed sums and Markov branching processes. Stoch. Process. Appl. 55, 285–300. Google Scholar
[33]Polito, F. and Scalas, E. (2016). A generalization of the space-fractional Poisson process and its connection to some Lévy processes. Electron. Commun. Prob. 21, 20. Google Scholar
[34]Pozdnyakov, V.et al. (2018). Discretely observed Brownian motion governed by telegraph process: estimation. Available at https://doi.org/10.1007/s11009-017-9547-6. Google Scholar
[35]Ratanov, N. (2013). Damped jump-telegraph processes. Statist. Prob. Lett. 83, 2282–2290. Google Scholar
[36]Ratanov, N. (2015). Telegraph processes with random jumps and complete market models. Methodol. Comput. Appl. Prob. 17, 677–695. Google Scholar
[37]Sandev, T., Tomovski, Z. and Crnkovic, B. (2017). Generalized distributed order diffusion equations with composite time fractional derivative. Comput. Math. Appl. 73, 1028–1040. Google Scholar