Advertisement

First Crossing Times of Telegraph Processes with Jumps

  • Nikita RatanovEmail author
Article
  • 18 Downloads

Abstract

The paper presents exact formulae related to the distribution of the first passage time τx of the jump-telegraph process. In particular, the Laplace transform of τx is analysed, when a jump component is in the opposite direction to the crossing level x > 0. The case of double exponential jumps is also studied in detail.

Keywords

Jump-telegraph process First passage time Laplace transformation Double exponential distribution 

Mathematics Subject Classification (2010)

60J75 60J27 60K99 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgments

I am very grateful to anonymous referees and the editor for the careful reading of the paper and for the helpful comments that have greatly improved the text.

References

  1. Abundo M (2000) On first-passage times for one-dimensional jump-diffusion processes. Probab Math Stat 20(2):399–423MathSciNetzbMATHGoogle Scholar
  2. Bogachev L, Ratanov N (2011) Occupation time distributions for the telegraph process. Stoch Process Appl 121:1816–1844MathSciNetCrossRefzbMATHGoogle Scholar
  3. Brémaud P (1999) Markov chains, Gibbs fields, Monte-Carlo simulation, and queues. Springer, BerlinzbMATHGoogle Scholar
  4. Di Crescenzo A, Iuliano A, Martinucci B, Zacks S (2013) Generalized telegraph process with random jumps. J Appl Probab 50(2):450–463MathSciNetCrossRefzbMATHGoogle Scholar
  5. Di Crescenzo A, Martinucci B (2013) On the generalized telegraph process with deterministic jumps. Methodol Comput Appl Probab 15(1):215–235MathSciNetCrossRefzbMATHGoogle Scholar
  6. Di Crescenzo A, Meoli A (2018) On a jump-telegraph process driven by an alternating fractional Poisson process. J Appl Probab 55(1):94–111MathSciNetCrossRefzbMATHGoogle Scholar
  7. Di Crescenzo A, Pellerey F (2002) On prices’ evolutions based on geometric telegrapher’s process. Appl Stoch Models Bus Ind 18:171–184MathSciNetCrossRefzbMATHGoogle Scholar
  8. Di Crescenzo A, Ratanov N (2015) On jump-diffusion processes with regime switching: martingale approach. ALEA Lat Am J Probab Math Stat 12(2):573–596MathSciNetzbMATHGoogle Scholar
  9. Fontbona J, Guérin H, Malrieu F (2016) Long time behavior of telegraph processes under convex potentials. Stoch Process Appl 126(10):3077–3101MathSciNetCrossRefzbMATHGoogle Scholar
  10. Foong SK (1992) First-passage time, maximum displacement, and Kac’s solution of the telegrapher equation. Phys Rev A 46:707–710MathSciNetCrossRefGoogle Scholar
  11. Foong SK, Kanno S (1994) Properties of the telegrapher’s random process with or without a trap. Stoch Process Appl 53:147–173MathSciNetCrossRefzbMATHGoogle Scholar
  12. Kolesnik AD, Ratanov N (2013) Telegraph processes and option pricing. Springer, HeidelbergCrossRefzbMATHGoogle Scholar
  13. Kou SG (2002) A jump-diffusion model for option pricing. Manag Sci 48 (8):1086–1101CrossRefzbMATHGoogle Scholar
  14. Kou SG, Wang H (2003) First passage times of a jump diffusion process. Adv Appl Probab 35:504–531MathSciNetCrossRefzbMATHGoogle Scholar
  15. López O, Ratanov N (2012) Kac’s rescaling for jump-telegraph processes. Statist Probab Lett 82:1768–1776MathSciNetCrossRefzbMATHGoogle Scholar
  16. López O, Ratanov N (2014) On the asymmetric telegraph processes. J Appl Probab 51:569–589MathSciNetCrossRefzbMATHGoogle Scholar
  17. Orsingher E (1990) Probability law, flow function, maximum distribution of wave-governed random motions and their connections with Kirchhoff’s laws. Stoch Process Appl 34:49–66MathSciNetCrossRefzbMATHGoogle Scholar
  18. Pogorui AA, Rodrguez-Dagnino RM, Kolomiets T (2015) The first passage time and estimation of the number of level-crossings for a telegraph process. Ukrain Math J 67(7):998–1007. (Ukrainian Original, 67(7):882–889)MathSciNetCrossRefzbMATHGoogle Scholar
  19. Prudnikov AP, Brychkov YuA, Marichev OI (1992) Integrals and series, vol 5. Inverse Laplace Transforms. Gordon and Breach Science PublGoogle Scholar
  20. Ratanov N (2007) A jump telegraph model for option pricing. Quant Finan 7:575–583MathSciNetCrossRefzbMATHGoogle Scholar
  21. Ratanov N (2010) Option pricing model based on a Markov-modulated diffusion with jumps. Braz J Probab Stat 24(2):413–431MathSciNetCrossRefzbMATHGoogle Scholar
  22. Ratanov N (2013) Damped jump-telegraph processes. Stat Probab Lett 83:2282–2290MathSciNetCrossRefzbMATHGoogle Scholar
  23. Ratanov N (2014) Double telegraph processes and complete market models. Stoch Anal App 32(4):555–574MathSciNetCrossRefzbMATHGoogle Scholar
  24. Ratanov N (2017) Self-exciting piecewise linear processes. ALEA Lat Am J Probab Math Stat 14:445–471MathSciNetzbMATHGoogle Scholar
  25. Ratanov N (2018) Kac-Lévy processes. J Theor Probab.  https://doi.org/10.1007/s10959-018-0873-6
  26. Shiryaev AN (2007) On martingale methods in the boundary crossing problems of Brownian motion. Sovrem Probl Mat 8:80. (in Russian)Google Scholar
  27. Zacks S (2004) Generalized integrated telegraph processes and the distribution of related stopping times. J Appl Probab 41(2):497–507MathSciNetCrossRefzbMATHGoogle Scholar
  28. Zacks S (2017) Sample path analysis and distributions of boundary crossing times. Lecture notes in mathematics, vol 2203. Springer, BerlinGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Universidad del RosarioBogotáColombia

Personalised recommendations