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Generalized Telegraph Process with Random Jumps

  • Antonio Di Crescenzo (a1), Antonella Iuliano (a1), Barbara Martinucci (a1) and Shelemyahu Zacks (a2)

Abstract

We consider a generalized telegraph process which follows an alternating renewal process and is subject to random jumps. More specifically, consider a particle at the origin of the real line at time t=0. Then it goes along two alternating velocities with opposite directions, and performs a random jump toward the alternating direction at each velocity reversal. We develop the distribution of the location of the particle at an arbitrary fixed time t, and study this distribution under the assumption of exponentially distributed alternating random times. The cases of jumps having exponential distributions with constant rates and with linearly increasing rates are treated in detail.

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Copyright

Corresponding author

Postal address: Dipartimento di Matematica, Università di Salerno, Via Giovanni Paolo II, n. 132, Fisciano (SA) 84084, Italy.
∗∗ Email address: adicrescenzo@unisa.it
∗∗∗ Email address: aiuliano@unisa.it
∗∗∗∗ Email address: bmartinucci@unisa.it
∗∗∗∗∗ Postal address: Department of Mathematical Sciences, Binghamton University, Binghamton, NY 13902-6000, USA. Email address: shelly@math.binghamton.edu

References

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[1] Beghin, L., Nieddu, L. and Orsingher, E. (2001). Probabilistic analysis of the telegrapher's process with drift by means of relativistic transformations. J. Appl. Math. Stoch. Anal. 14, 1125.
[2] Boxma, O., Perry, D., Stadje, W. and Zacks, S. (2006). A Markovian growth-collapse model. Adv. Appl. Prob. 38, 221243.
[3] Bshouty, D., Di Crescenzo, A., Martinucci, B. and Zacks, S. (2012). Generalized telegraph process with random delays. J. Appl. Prob. 49, 850865.
[4] De Gregorio, A. and Iacus, S. M. (2008). Parametric estimation for standard and geometric telegraph process observed at discrete times. Statist. Infer. Stoch. Process. 11, 249263.
[5] De Gregorio, A. and Iacus, S. M. (2011). Least-squares change-point estimation for the telegraph process observed at discrete times. Statistics 45, 349359.
[6] Di Crescenzo, A. (2001). On random motions with velocities alternating at Erlang-distributed random times. Adv. Appl. Prob. 33, 690701.
[7] Di Crescenzo, A. and Martinucci, B. (2010). A damped telegraph random process with logistic stationary distribution. J. Appl. Prob. 47, 8496.
[8] Di Crescenzo, A. and Martinucci, B. (2013). On the generalized telegraph process with deterministic Jumps. Methodology Comput. Appl. Prob. 15, 215235.
[9] Di Crescenzo, A. and Pellerey, F. (2002). On prices' evolutions based on geometric telegrapher's process. Appl. Stoch. Models Business Industry 18, 171184.
[10] Di Crescenzo, A., Martinucci, B. and Zacks, S. (2012). On the damped geometric telegrapher's process. In Mathematical and Statistical Methods for Actuarial Sciences and Finance, eds Perna, C. and Sibillo, M.. Springer, Dordrecht, pp. 175182.
[11] Goldstein, S. (1951). On diffusion by discontinuous movements, and on the telegraph equation. Quart. J. Mech. Appl. Math. 4, 129156.
[12] Gradshteyn, I. S. and Ryzhik, I. M. (2007). Tables of Integrals, Series, and Products, 7th edn. Academic Press, Amsterdam.
[13] Kac, M. (1974). A stochastic model related to the telegrapher's equation. Rochy Mountain J. Math. 4, 497509.
[14] López, O. and Ratanov, N. (2012). Kac's rescaling for Jump-telegraph processes. Statist. Prob. Lett. 82, 17681776.
[15] López, O. and Ratanov, N. (2012). Option pricing driven by a telegraph process with random Jumps. J. Appl. Prob. 49, 838849.
[16] Mazza, C. and Rullière, D. (2004). A link between wave governed random motions and ruin processes. Insurance Math. Econom. 35, 205222.
[17] 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, 4966.
[18] Perry, D., Stadje, W. and Zacks, S. (2005). A two-sided first-exit problem for a compound Poisson process with a random upper boundary. Methodology Comput. Appl. Prob. 7, 5162.
[19] Pinsky, M. A. (1991). Lectures on Random Evolutions. World Scientific, River Edge, NJ.
[20] Ratanov, N. (2007). A Jump telegraph model for option pricing. Quant. Finance 7, 575583.
[21] Ratanov, N. (2007). Jump telegraph processes and financial markets with memory. J. Appl. Math. Stoch. Anal. 2007, 19pp.
[22] Ratanov, N. (2010). Option pricing model based on a Markov-modulated diffusion with Jumps. Braz. J. Prob. Statist. 24, 413431.
[23] Ratanov, N. and Melnikov, A. (2008). On financial markets based on telegraph processes. Stochastics 80, 247268.
[24] Stadje, W. and Zacks, S. (2004). Telegraph processes with random velocities. J. Appl. Prob. 41, 665678.
[25] Van Lieshout, M. N. M. (2006). Maximum likelihood estimation for random sequential adsorption. Adv. Appl. Prob. 38, 889898.
[26] Zacks, S. (2004). Generalized integrated telegrapher process and the distribution of related stopping times. J. Appl. Prob. 41, 497507.

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