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Journal of Applied Mathematics and Stochastic Analysis
Volume 2007, Article ID 72326, 19 pages
http://dx.doi.org/10.1155/2007/72326
Research Article

Jump Telegraph Processes and Financial Markets with Memory

Faculty of Economics, Universidad del Rosario, Calle 14, No.4-69, Bogotá, Colombia

Received 21 November 2006; Revised 22 April 2007; Accepted 9 August 2007

Copyright © 2007 Nikita Ratanov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. T. Björk and H. Hult, “A note on Wick products and the fractional Black-Scholes model,” Finance and Stochastics, vol. 9, no. 2, pp. 197–209, 2005. View at Zentralblatt MATH · View at MathSciNet
  2. P. Carr, H. German, D. B. Madan, and M. Yor, “The fine structure of asset returns: an empirical investigation,” The Journal of Business, vol. 75, no. 2, pp. 305–332, 2002. View at Publisher · View at Google Scholar
  3. P. Carr, H. Geman, D. B. Madan, and M. Yor, “Pricing options on realized variance,” Finance and Stochastics, vol. 9, no. 4, pp. 453–475, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. R. J. Elliott and C.-J. U. Osakwe, “Option pricing for pure jump processes with Markov switching compensators,” Finance and Stochastics, vol. 10, no. 2, pp. 250–275, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. S. Goldstein, “On diffusion by discontinuous movements and on the telegraph equation,” The Quarterly Journal of Mechanics and Applied Mathematics, vol. 4, no. 2, pp. 129–156, 1951. View at Zentralblatt MATH · View at MathSciNet
  6. M. Kac, “A stochastic model related to the telegrapher's equation,” The Rocky Mountain Journal of Mathematics, vol. 4, pp. 497–509, 1974. View at Zentralblatt MATH · View at MathSciNet
  7. E. Orsingher, “Probability law, flow function, maximum distribution of wave-governed random motions and their connections with Kirchoff's laws,” Stochastic Processes and their Applications, vol. 34, no. 1, pp. 49–66, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. S. Zacks, “Generalized integrated telegraph processes and the distribution of related stopping times,” Journal of Applied Probability, vol. 41, no. 2, pp. 497–507, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. G. B. Di Mazi, Y. M. Kabanov, and V. I. Runggal'der, “Mean-variance hedging of options on stocks with Markov volatilities,” Theory of Probability and Its Applications, vol. 39, no. 1, pp. 172–182, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. C. Mazza and D. Rullière, “A link between wave governed random motions and ruin processes,” Insurance: Mathematics & Economics, vol. 35, no. 2, pp. 205–222, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. A. Di Crescenzo and F. Pellerey, “On prices' evolutions based on geometric telegrapher's process,” Applied Stochastic Models in Business and Industry, vol. 18, no. 2, pp. 171–184, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. M. Kac, Probability and Related Topics in Physical Sciences, vol. 1 of Lectures in Applied Mathematics, Interscience Publishers, London, UK, 1959. View at Zentralblatt MATH · View at MathSciNet
  13. P. Protter, Stochastic Integration and Differential Equations. A New Approach, vol. 21 of Applications of Mathematics, Springer, Berlin, Germany, 1990. View at Zentralblatt MATH · View at MathSciNet
  14. L. Beghin, L. Nieddu, and E. Orsingher, “Probabilistic analysis of the telegrapher's process with drift by means of relativistic transformations,” Journal of Applied Mathematics and Stochastic Analysis, vol. 14, no. 1, pp. 11–25, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. I. Karatzas and S. E. Shreve, Methods of Mathematical Finance, vol. 39 of Applications of Mathematics, Springer, New York, NY, USA, 1998. View at Zentralblatt MATH · View at MathSciNet
  16. N. Ratanov, “A jump telegraph model for option pricing,” Quantitative Finance, vol. 7, no. 5, pp. 575–583, 2007.
  17. N. Ratanov, “Pricing options under telegraph processes,” Revista de Economia del Rosario, vol. 8, no. 2, pp. 131–150, 2005.
  18. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, NY, USA, 1972.
  19. N. Ratanov, “Telegraph evolutions in inhomogeneous media,” Markov Processes and Related Fields, vol. 5, no. 1, pp. 53–68, 1999. View at Zentralblatt MATH · View at MathSciNet
  20. V. Anh and A. Inoue, “Financial markets with memory. I. Dynamic models,” Stochastic Analysis and Applications, vol. 23, no. 2, pp. 275–300, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. A. Inoue, Y. Nakano, and V. Anh, “Linear filtering of systems with memory and application to finance,” Journal of Applied Mathematics and Stochastic Analysis, vol. 2006, Article ID 53104, 26 pages, 2006. View at Publisher · View at Google Scholar · View at MathSciNet