Skip to main content
Log in

On the Generalized Telegraph Process with Deterministic Jumps

  • Published:
Methodology and Computing in Applied Probability Aims and scope Submit manuscript

Abstract

We consider a semi-Markovian generalization of the integrated telegraph process subject to jumps. It describes a motion on the real line characterized by two alternating velocities with opposite directions, where a jump along the alternating direction occurs at each velocity reversal. We obtain the formal expressions of the forward and backward transition densities of the motion. We express them as series in the case of Erlang-distributed random times separating consecutive jumps. Furthermore, a closed form of the transition density is given for exponentially distributed times, with constant jumps and random initial velocity. In this case we also provide mean and variance of the process, and study the limiting behaviour of the probability law, which leads to a mixture of three Gaussian densities.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Abramowitz M, Stegun IA (1992) Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover, New York

    Google Scholar 

  • Beghin L, Nieddu L, Orsingher E (2001) Probabilistic analysis of the telegrapher’s process with drift by means of relativistic transformations. J Appl Math Stoch Anal 14:11–25

    Article  MathSciNet  MATH  Google Scholar 

  • Boxma O, Perry D, Stadje W, Zacks S (2006) A Markovian growth-collapse model. Adv Appl Probab 38:221–243

    Article  MathSciNet  MATH  Google Scholar 

  • Di Crescenzo A (2001) On random motions with velocities alternating at Erlang-distributed random times. Adv Appl Probab 33:690–701

    Article  MATH  Google Scholar 

  • Di Crescenzo A, Martinucci B (2010) A damped telegraph random process with logistic stationary distribution. J Appl Probab 47:84–96

    Article  MathSciNet  MATH  Google Scholar 

  • Di Crescenzo A, Pellerey F (2002) On prices’ evolutions based on geometric telegrapher’s process. Appl Stoch Models Bus Ind 18:171–184

    Article  MathSciNet  MATH  Google Scholar 

  • Goldstein S (1951) On diffusion by discontinuous movements and the telegraph equation. Q J Mech Appl Math 4:129–156

    Article  MATH  Google Scholar 

  • Kac M (1974) A stochastic model related to the telegrapher’s equation. Rochy Mountain J Math 4:497–509

    Article  MATH  Google Scholar 

  • Lachal A (2006) Cyclic random motions in ℝd-space with n directions. ESAIM Probab Stat 10:277–316

    Article  MathSciNet  MATH  Google Scholar 

  • Mazza C, Rullière D (2004) A link between wave governed random motions and ruin processes. Insur, Math Econ 35:205–222

    Article  MATH  Google Scholar 

  • Orsingher E (1990a) Probability law, flow function, maximum distribution of wave-governed random motions and their connections with Kirchoff’s laws. Stoch Process Their Appl 34:49–66

    Article  MathSciNet  MATH  Google Scholar 

  • Orsingher E (1990b) Random motions governed by third-order equations. Adv Appl Probab 22:915–928

    Article  MathSciNet  MATH  Google Scholar 

  • Pellerey F, Shaked M (1993) Stochastic comparison of some wear processes. Probab Eng Inf Sci 7:421–435

    Article  Google Scholar 

  • Pellerey F, Shaked M (1996) Stochastic comparison of processes generated by random interruptions of monotone functions and related results. Lifetime Data Anal 2:91–112

    Article  MATH  Google Scholar 

  • Perry D, Stadje W, Zacks S (2002) First-exit times for compound Poisson processes for some types of positive and negative jumps. Stoch Models 18:139–157

    Article  MathSciNet  MATH  Google Scholar 

  • Perry D, Stadje W, Zacks S (2005) A two-sided first-exit problem for a compound Poisson process with a random upper boundary. Methodol Comput Appl Probab 7:51–62

    Article  MathSciNet  MATH  Google Scholar 

  • Ratanov N (2007a) A jump telegraph model for option pricing. Quantitative Finance 7:575–583

    Article  MathSciNet  MATH  Google Scholar 

  • Ratanov N (2007b) Jump telegraph processes and financial markets with memory. J Appl Math Stoch Anal. doi:10.1155/2007/72326

    MathSciNet  Google Scholar 

  • Ratanov N, Melnikov A (2008) On financial markets based on telegraph processes. Stochastics 80:247–268

    MathSciNet  MATH  Google Scholar 

  • Stadje W, Zacks S (2004) Telegraph processes with random velocities. J Appl Probab 41:665–678

    Article  MathSciNet  MATH  Google Scholar 

  • Zacks S (2004) Generalized integrated telegrapher process and the distribution of related stopping times. J Appl Probab 41:497–507

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Antonio Di Crescenzo.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Di Crescenzo, A., Martinucci, B. On the Generalized Telegraph Process with Deterministic Jumps. Methodol Comput Appl Probab 15, 215–235 (2013). https://doi.org/10.1007/s11009-011-9235-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11009-011-9235-x

Keywords

AMS 2000 Subject Classification

Navigation