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

  • Antonio Di Crescenzo (a1) and Alessandra Meoli (a1)

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.

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Corresponding author

* Postal address: Dipartimento di Matematica, Università di Salerno, Via Giovanni Paolo II, 132, 84084 Fisciano, SA, Italy.
** Email address: adicrescenzo@unisa.it
*** Email address: ameoli@unisa.it

References

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[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, 187204.
[2]Beghin, L. and Orsingher, E. (2009). Fractional Poisson processes and related planar random motions. Electron. J. Prob. 14, 17901827.
[3]Beghin, L. and Orsingher, E. (2010). Poisson-type processes governed by fractional and higher-order recursive differential equations. Electron. J. Prob. 15, 684709.
[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. 317.
[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, 249263.
[6]De Gregorio, A. and Iacus, S. M. (2011). Least-squares change-point estimation for the telegraph process observed at discrete times. Statistics 45, 349359.
[7]Di Crescenzo, A. (2001). On random motions with velocities alternating at Erlang-distributed random times. Adv. Appl. Prob. 33, 690701.
[8]Di Crescenzo, A. and Martinucci, B. (2010). A damped telegraph random process with logistic stationary distribution. J. Appl. Prob. 47, 8496.
[9]Di Crescenzo, A. and Martinucci, B. (2013). On the generalized telegraph process with deterministic jumps. Methodol. Comput. Appl. Prob. 15, 215235.
[10]Di Crescenzo, A. and Meoli, A. (2016). On a fractional alternating Poisson process. AIMS Math. 1, 212224.
[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, 291307.
[12]Di Crescenzo, A., Iuliano, A., Martinucci, B. and Zacks, S. (2013). Generalized telegraph process with random jumps. J. Appl. Prob. 50, 450463.
[13]Ferraro, S., Manzini, M., Masoero, A. and Scalas, E. (2009). A random telegraph signal of Mittag-Leffler type. Physica A 388, 39913999.
[14]Garcia, R.et al. (2007). Optimal foraging by zooplankton within patches: the case of Daphnia. Math. Biosci. 207, 165188.
[15]Goldstein, S. (1951). On diffusion by discontinuous movements, and on the telegraph equation. Quart. J. Mech. Appl. Math. 4, 129156.
[16]Gorenflo, R., Kilbas, A. A., Mainardi, F. and Rogosin, S. V. (2014). Mittag-Leffler Functions, Related Topics and Applications. Springer, Heidelberg.
[17]Iacus, S. M. and Yoshida, N. (2009). Estimation for the discretely observed telegraph process. Theory Prob. Math. Statist. 78, 3747.
[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. 7992.
[19]Kac, M. (1974). A stochastic model related to the telegrapher's equation. Rocky Mountain J. Math. 4, 497509.
[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, 3149.
[21]Kolesnik, A. D. and Ratanov, N. (2013). Telegraph Processes and Option Pricing. Springer, Heidelberg.
[22]López, O. and Ratanov, N. (2012). Kac's rescaling for jump-telegraph processes. Statist. Prob. Lett. 82, 17681776.
[23]López, O. and Ratanov, N. (2014). On the asymmetric telegraph processes. J. Appl. Prob. 51, 569589.
[24]Masoliver, J. (2016). Fractional telegrapher's equation from fractional persistent random walks. Phys. Rev. E 93, 052107.
[25]Mathai, A. M. and Haubold, H. J. (2008). Special Functions for Applied Scientists. Springer, New York.
[26]Orsingher, E. and Beghin, L. (2004). Time-fractional telegraph equations and telegraph processes with Brownian time. Prob. Theory Relat. Fields 128, 141160.
[27]Orsingher, E. and Polito, F. (2010). Fractional pure birth processes. Bernoulli 16, 858881.
[28]Orsingher, E. and Polito, F. (2011). On a fractional linear birth-death process. Bernoulli 17, 114137.
[29]Orsingher, E. and Zhao, X. (2003). The space-fractional telegraph equation and the related fractional telegraph process. Chinese Ann. Math. B 24, 4556.
[30]Orsingher, E., Polito, F. and Sakhno, L. (2010). Fractional non-linear, linear and sublinear death processes. J. Statist. Phys. 141, 6893.
[31]Othmer, H. G., Dunbar, S. R. and Alt, W. (1988). Models of dispersal in biological systems. J. Math. Biol. 26, 263298.
[32]Pakes, A. G. (1995). Characterization of discrete laws via mixed sums and Markov branching processes. Stoch. Process. Appl. 55, 285300.
[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.
[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.
[35]Ratanov, N. (2013). Damped jump-telegraph processes. Statist. Prob. Lett. 83, 22822290.
[36]Ratanov, N. (2015). Telegraph processes with random jumps and complete market models. Methodol. Comput. Appl. Prob. 17, 677695.
[37]Sandev, T., Tomovski, Z. and Crnkovic, B. (2017). Generalized distributed order diffusion equations with composite time fractional derivative. Comput. Math. Appl. 73, 10281040.

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

  • Antonio Di Crescenzo (a1) and Alessandra Meoli (a1)

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