On the Laws of First Hitting Times of Points for One-Dimensional Symmetric Stable Lévy Processes

  • Kouji YanoEmail author
  • Yuko Yano
  • Marc Yor
Part of the Lecture Notes in Mathematics book series (LNM, volume 1979)


Several aspects of the laws of first hitting times of points are investigated for one-dimensional symmetric stable Lévy processes. Itô’s excursion theory plays a key role in this study.


Symmetric stable Lévy process excursion theory first hitting times 


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  1. 1.
    O. Barndorff-Nielsen, J. Kent, and M. Sørensen. Normal variance-mean mixtures and z distributions. Internat. Statist. Rev., 50(2):145–159, 1982.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    J. Bertoin. Lévy processes, volume 121 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1996.zbMATHGoogle Scholar
  3. 3.
    J. Bertoin, T. Fujita, B. Roynette, and M. Yor. On a particular class of self-decomposable random variables: the durations of Bessel excursions straddling independent exponential times. Probab. Math. Statist., 26(2):315–366, 2006.MathSciNetzbMATHGoogle Scholar
  4. 4.
    R. M. Blumenthal and R. K. Getoor. Markov processes and potential theory. Pure and Applied Mathematics, Vol. 29. Academic Press, New York, 1968.Google Scholar
  5. 5.
    R. M. Blumenthal, R. K. Getoor, and D. B. Ray. On the distribution of first hits for the symmetric stable processes. Trans. Amer. Math. Soc., 99:540–554, 1961.MathSciNetzbMATHGoogle Scholar
  6. 6.
    L. Bondesson. On the infinite divisibility of the half-Cauchy and other decreasing densities and probability functions on the nonnegative line. Scand. Actuar. J., (3–4):225–247, 1987.MathSciNetCrossRefGoogle Scholar
  7. 7.
    L. Bondesson. Generalized gamma convolutions and related classes of distributions and densities, volume 76 of Lecture Notes in Statistics. Springer-Verlag, New York, 1992.zbMATHCrossRefGoogle Scholar
  8. 8.
    P. Bourgade, T. Fujita, and M. Yor. Euler's formulae for ζ(2n) and products of Cauchy variables. Electron. Comm. Probab., 12:73–80 (electronic), 2007.MathSciNetzbMATHGoogle Scholar
  9. 9.
    J. Bretagnolle. Résultats de Kesten sur les processus à accroissements indépendants. In Séminaire de Probabilités, V (Univ. Strasbourg, année universitaire 1969–1970), pages 21–36. Lecture Notes in Math., Vol. 191. Springer, Berlin, 1971.CrossRefGoogle Scholar
  10. 10.
    P. Carmona, F. Petit, and M. Yor. On the distribution and asymptotic results for exponential functionals of Lévy processes. In Exponential functionals and principal values related to Brownian motion, Bibl. Rev. Mat. Iberoamericana, pages 73–130. Rev. Mat. Iberoamericana, Madrid, 1997.Google Scholar
  11. 11.
    L. Chaumont and M. Yor. Exercises in probability, A guided tour from measure theory to random processes, via conditioning, volume 13 of Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2003.CrossRefGoogle Scholar
  12. 12.
    L. Devroye. A note on Linnik's distribution. Statist. Probab. Lett., 9(4):305–306, 1990.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    C. Donati-Martin, B. Roynette, P. Vallois, and M. Yor. On constants related to the choice of the local time at 0, and the corresponding Itâ measure for Bessel processes with dimension d = 2(1 − α), 0 < α < 1. Studia Sci. Math. Hungar., 45(2):207–221, 2008.MathSciNetzbMATHGoogle Scholar
  14. 14.
    W. Feller. An introduction to probability theory and its applications. Vol. II. Second edition. John Wiley & Sons Inc., New York, 1971.zbMATHGoogle Scholar
  15. 15.
    T. Fujita, Y. Yano, and M. Yor. in preparation.Google Scholar
  16. 16.
    R. K. Getoor. Continuous additive functionals of a Markov process with applications to processes with independent increments. J. Math. Anal. Appl., 13:132–153, 1966.MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    B. Grigelionis. Processes of Meixner type. Liet. Mat. Rink., 39(1):40–51, 1999.MathSciNetGoogle Scholar
  18. 18.
    B. Grigelionis. Generalized z-distributions and related stochastic processes. Liet. Mat. Rink., 41(3):303–319, 2001.MathSciNetGoogle Scholar
  19. 19.
    B. Grigelionis. On the self-decomposability of Euler's gamma function. Liet. Mat. Rink., 43(3):359–370, 2003.MathSciNetGoogle Scholar
  20. 20.
    M. Hayashi and K. Yano. On the laws of total local times for h-paths of stable Lévy processes. in preparation.Google Scholar
  21. 21.
    I. A. Ibragimov and Yu. V. Linnik. Independent and stationary sequences of random variables. Wolters-Noordhoff Publishing, Groningen, 1971. With a supplementary chapter by I. A. Ibragimov and V. V. Petrov, Translation from the Russian edited by J. F. C. Kingman.zbMATHGoogle Scholar
  22. 22.
    N. Ikeda and S. Watanabe. Stochastic differential equations and diffusion processes, volume 24 of North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam, second edition, 1989.zbMATHGoogle Scholar
  23. 23.
    K. Itâ. Poisson point processes attached to Markov processes. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. III: Probability theory, pages 225–239, Berkeley, Calif., 1972. Univ. California Press.Google Scholar
  24. 24.
    K. Itâ and H. P. McKean Jr., Diffusion processes and their sample paths. Springer-Verlag, Berlin, 1974. Second printing, corrected, Die Grundlehren der mathematischen Wissenschaften, Band 125.Google Scholar
  25. 25.
    L. F. James. Gamma tilting calculus for GGC and Dirichlet means with applications to Linnik processes and occupation time laws for randomly skewed Bessel processes and bridges. preprint, arXiv:math/0610218, 2006.Google Scholar
  26. 26.
    L. F. James, B. Roynette, and M. Yor. Generalized Gamma convolutions, Dirichlet means, Thorin measures with explicit examples. Probab. Surv., 5:346–415, 2008.MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    H. Kesten. Hitting probabilities of single points for processes with stationary independent increments. Memoirs of the American Mathematical Society, No. 93. American Mathematical Society, Providence, R.I., 1969.Google Scholar
  28. 28.
    F. B. Knight. Brownian local times and taboo processes. Trans. Amer. Math. Soc., 143:173–185, 1969.MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    M. B. Marcus and J. Rosen. Markov processes, Gaussian processes, and local times, volume 100 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2006.zbMATHCrossRefGoogle Scholar
  30. 30.
    P. A. Meyer. Processus de Poisson ponctuels, d'après K. Ito. In Séminaire de Probabilités, V (Univ. Strasbourg, année universitaire 1969–1970), pages 177–190. Lecture Notes in Math., Vol. 191. Springer, Berlin, 1971.CrossRefGoogle Scholar
  31. 31.
    J. Pitman and M. Yor. Bessel processes and infinitely divisible laws. In Stochastic integrals (Proc. Sympos., Univ. Durham, Durham, 1980), volume 851 of Lecture Notes in Math., pages 285–370. Springer, Berlin, 1981.Google Scholar
  32. 32.
    J. Pitman and M. Yor. Infinitely divisible laws associated with hyperbolic functions. Canad. J. Math., 55(2):292–330, 2003.MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    J. Pitman and M. Yor. Itâ's excursion theory and its applications. Jpn. J. Math., 2(1):83–96, 2007.MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    D. Ray. Stable processes with an absorbing barrier. Trans. Amer. Math. Soc., 89:16–24, 1958.MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    D. Revuz and M. Yor. Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin, third edition, 1999.zbMATHGoogle Scholar
  36. 36.
    B. Roynette, P. Vallois, and M. Yor. A family of generalized gamma convoluted variables. to appear in Prob. Math. Stat., 2009.Google Scholar
  37. 37.
    P. Salminen. On last exit decompositions of linear diffusions. Studia Sci. Math. Hungar., 33(1–3):251–262, 1997.MathSciNetzbMATHGoogle Scholar
  38. 38.
    W. Schoutens. Stochastic processes and orthogonal polynomials, volume 146 of Lecture Notes in Statistics. Springer-Verlag, New York, 2000.zbMATHCrossRefGoogle Scholar
  39. 39.
    W. Schoutens. Lévy processes in finance: Pricing financial derivatives. John Wiley & Sons Inc., 2003.Google Scholar
  40. 40.
    W. Schoutens and J. L. Teugels. Lévy processes, polynomials and martingales. Comm. Statist. Stochastic Models, 14(1–2):335–349, 1998. Special issue in honor of Marcel F. Neuts.MathSciNetzbMATHGoogle Scholar
  41. 41.
    D. N. Shanbhag and M. Sreehari. On certain self-decomposable distributions. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 38(3):217–222, 1977.MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    O. Thorin. On the infinite divisibility of the Pareto distribution. Scand. Actuar. J., (1):31–40, 1977.Google Scholar
  43. 43.
    D. Williams. Decomposing the Brownian path. Bull. Amer. Math. Soc., 76:871–873, 1970.MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    D. Williams. Path decomposition and continuity of local time for one-dimensional diffusions. I. Proc. London Math. Soc. (3), 28:738–768, 1974.CrossRefGoogle Scholar
  45. 45.
    M. Winkel. Electronic foreign-exchange markets and passage events of independent subordinators. J. Appl. Probab., 42(1):138–152, 2005.MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    M. Yamazato. Topics related to gamma processes. In Stochastic processes and applications to mathematical finance, pages 157–182. World Sci. Publ., Hackensack, NJ, 2006.Google Scholar
  47. 47.
    K. Yano. Convergence of excursion point processes and its applications to functional limit theorems of markov processes on a half line. Bernoulli, 14(4):963–987, 2008.MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    K. Yano. Excursions away from a regular point for one-dimensional symmetric Lévy processes without Gaussian part. submitted. preprint, arXiv:0805.3881, 2008.Google Scholar
  49. 49.
    K. Yano, Y. Yano, and M. Yor. Penalising symmetric stable Lévy paths. J. Math. Soc. Japan, to appear in 2009.Google Scholar
  50. 50.
    F. Cordero. Sur la théorie des excursions des processus de Lévy et quelques applications. in preparation.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.Department of MathematicsGraduate School of Science, Kobe UniversityKobeJapan
  2. 2.Research Institute for Mathematical Sciences, Kyoto UniversityKyotoJapan
  3. 3.Laboratoire de Probabilités et Modéles Aléatoires, Université Paris VIParisFrance
  4. 4.Institut Universitaire de FranceFrance

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