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Piecewise deterministic processes following two alternating patterns

Part of: Special processes Markov processes

Published online by Cambridge University Press:  11 December 2019

Nikita Ratanov ,
Antonio Di Crescenzo  and
Barbara Martinucci
Nikita Ratanov*
Affiliation:
Universidad del Rosario
Antonio Di Crescenzo*
Affiliation:
Università di Salerno
Barbara Martinucci*
Affiliation:
Università di Salerno
*
* Postal address: Facultad de Economía, Universidad del Rosario, Calle 12c, No. 4-69, Bogotá, D. C. Cundinamarca, Colombia.
*** Postal address: Dipartimento di Matematica, Università di Salerno, 84084 Fisciano (SA), Italia.
*** Postal address: Dipartimento di Matematica, Università di Salerno, 84084 Fisciano (SA), Italia.
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Abstract

We propose a wide generalization of known results related to the telegraph process. Functionals of the simple telegraph process on a straight line and their generalizations on an arbitrary state space are studied.

Keywords

Telegraph processrectifying diffeomorphism

MSC classification

Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 60K15: Markov renewal processes, semi-Markov processes 60K99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 4 , December 2019 , pp. 1006 - 1019
Copyright
© Applied Probability Trust 2019 

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References

Arnold, V. I. (1992). Ordinary Differential Equations. Springer, Berlin.Google Scholar
Bogachev, L. and Ratanov, N. (2011). Occupation time distributions for the telegraph process. Stoch. Process. Appl. 121, 1816–1844.CrossRefGoogle Scholar
Boxma, O., Kaspi, H., Kella, O. and Perry, D. (2005). On/off storage systems with state-dependent input, output and switching rates. Probab. Engrg. Inform. Sci. 19, 1–14.CrossRefGoogle Scholar
Davis, M. H. A. (1984). Piecewise-deterministic Markov processes: A general class of non-diffusion stochastic models. J. R. Statist. Soc. B, 46, 353–388.Google Scholar
Davis, M. H. A. (1993). Markov Models and Optimization. Chapman & Hall, London.CrossRefGoogle Scholar
De Gregorio, A. (2010). Stochastic velocity motions and processes with random time. Adv. Appl. Prob. 42, 1028–1056.CrossRefGoogle Scholar
Di Crescenzo, A. and Martinucci, B. (2010). A damped telegraph random process with logistic stationary distribution. J. Appl. Prob. 47, 84–96.CrossRefGoogle Scholar
Embrechts, P. and Schmidli, H (1994). Ruin estimation for a general insurance risk model. Adv. Appl. Prob. 26, 404–422.CrossRefGoogle Scholar
Garra, R., Orsingher, E. and Ratanov, N. (2017). Planar piecewise linear random motions with jumps. Math. Meth. Appl. Sci. 40, 7673–7685.CrossRefGoogle Scholar
Gnedenko, B. V. and Kovalenko, I. N. (1989). Introduction to Queueing Theory, 2nd edn. Birkhaüser, Boston.Google Scholar
Jacobsen, M. (2006). Point Process Theory and Applications. Marked Point and Piecewise Deterministic Processes. Birkhäuser, Boston.Google Scholar
Kac, M. (1974). A stochastic model related to the telegrapher’s equation. Rocky Mountain J. Math. 4, 497–509.CrossRefGoogle Scholar
Kolesnik, A. D. (1998). The equations of Markovian random evolution on the line. J. Appl. Prob. 35, 27–35.CrossRefGoogle Scholar
Kolesnik, A. D. (2007). A note on planar random motion at finite speed. J. Appl. Prob. 44, 838–842.CrossRefGoogle Scholar
Kolesnik, A. D. and Orsingher, E. (2005) A planar random motion with an infinite number of directions controlled by the damped wave equation. J. Appl. Prob. 42, b 1168–1182.CrossRefGoogle Scholar
Kolesnik, A. D. and Ratanov, N. (2013). Telegraph Processes and Option Pricing. Springer, Heidelberg.CrossRefGoogle Scholar
Kyprianou, A. E. (2014). Fluctuations of Lévy Processes with Applications. Introductory lectures, 2nd edn. Springer, Heidelberg.CrossRefGoogle Scholar
Lasota, A., Mackey, M. C. and Tyrcha, J. (1992). The statistical dynamics of recurrent biological events. J. Math. Biol. 30, 775–800.CrossRefGoogle Scholar
López, O. and Ratanov, N. (2012). Option pricing driven by a telegraph process with random jumps. J. Appl. Prob. 49, 838–849.CrossRefGoogle Scholar
Martinucci, B. and Meoli, A. (2019). On certain functionals of squared telegraph processes. Stoch. Dynam. Available at https://doi.org/10.1142/S0219493720500057.CrossRefGoogle Scholar
Orsingher, E. (2000). Exact joint distribution in a model of planar random motion. Stoch. Stoch. Rep. 69, 1–10.CrossRefGoogle Scholar
Orsingher, E. and Ratanov, N. (2002) Planar random motions with drift. J. Appl. Math. Stoch. Anal. 15, 189–205.CrossRefGoogle Scholar
Ratanov, N. (2007). Jump telegraph processes and financial markets with memory. J. Appl. Math. Stoch. Anal. 2007, 72326.CrossRefGoogle Scholar
Stadje, W. and Zacks, S. (2004). Telegraph processes with random velocities. J. Appl. Prob. 41, 665–678.CrossRefGoogle Scholar