Large Deviations for Clocks of Self-similar Processes

Abstract

The Lamperti correspondence gives a prominent role to two random time changes: the exponential functional of a Lévy process drifting to \(\infty\) and its inverse, the clock of the corresponding positive self-similar process. We describe here asymptotical properties of these clocks in large time, extending the results of Yor and Zani (Bernoulli 7, 351–362, 2001).

References

  1. 1.
    J. Bertoin, Random Fragmentation and Coagulation Processes. Cambridge Studies in Advanced Mathematics, vol. 102 (Cambridge University Press, Cambridge, 2006)Google Scholar
  2. 2.
    J. Bertoin, M.E. Caballero, Entrance from 0+ for increasing semi-stable Markov processes. Bernoulli 8(2), 195–205 (2002)MATHMathSciNetGoogle Scholar
  3. 3.
    J. Bertoin, M. Yor, The entrance laws of self-similar Markov processes and exponential functionals of Lévy processes. Potential Anal. 17(4), 389–400 (2002)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    J. Bertoin, M. Yor, Exponential functionals of Lévy processes. Probab. Surv. 2, 191–212 (2005)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    M.E. Caballero, V. Rivero, On the asymptotic behaviour of increasing self-similar Markov processes. Electron. J. Probab. 14, 865–894 (2009)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    M.E. Caballero, J.C. Pardo, J.L. Pérez, Explicit identities for Lévy processes associated to symmetric stable processes. Bernoulli 17(1), 34–59 (2011)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Ph. Carmona, F. Petit, M. Yor, On exponential functionals of certain Lévy processes. Stochast. Stochast. Rep. 47, 71–101 (1994)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Ph. Carmona, F. Petit, 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, (Rev. Mat. Iberoamericana, Madrid), pp. 73–130 (1997)Google Scholar
  9. 9.
    Ph. Carmona, F. Petit, M. Yor, Beta-gamma random variables and intertwining relations between certain Markov processes. Rev. Matemática Iberoam. 14(2), 311–367 (1998)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Ph. Carmona, F. Petit, M. Yor, Exponential functionals of Lévy processes, in Lévy Processes (Birkhauser, Boston, 2001), pp. 41–55Google Scholar
  11. 11.
    A. Dembo, O. Zeitouni, Large Deviations Techniques and Applications (Springer, New York, 1998)MATHCrossRefGoogle Scholar
  12. 12.
    N. Demni, M. Zani, Large deviations for statistics of the Jacobi process. Stochast. Process. Appl. 119(2), 518–533 (2009)MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    N.G. Duffield, W. Whitt, Large deviations of inverse processes with nonlinear scalings. Ann. Appl. Probab. 8(4), 995–1026 (1998)MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    U. Küchler, M. Sørensen, Exponential Families of Stochastic Processes. Springer Series in Statistics (Springer, New York, 1997)Google Scholar
  15. 15.
    A. Kyprianou, Fluctuations of Lévy Processes with Applications. Universitext, 2nd edn. (Springer, Heidelberg, 2014) [Introductory lectures]Google Scholar
  16. 16.
    A.E. Kyprianou, J.C. Pardo, A.R. Watson, The extended hypergeometric class of Lévy processes. J. Appl. Probab. 51A, 391–408 (2014)MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    J. Lamperti, Semi-stable Markov processes I. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 22, 205–225 (1972)MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    G. Letac, M. Mora, Natural exponential families with cubic variance. Ann. Stat. 18, 1–37 (1990)MATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    R. Mansuy, M. Yor, Aspects of Brownian Motion. Universitext (Springer, Berlin, 2008)MATHCrossRefGoogle Scholar
  20. 20.
    K. Maulik, B. Zwart, Tail asymptotics for exponential of Lévy processes. Stoch. Process. Appl. 116, 156–177 (2006)MATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    P. Patie, Exponential functional of a new family of Lévy processes and self-similar continuous state branching processes with immigration. Bull. Sci. Math. 133(4), 355–382 (2009)MATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    D. Revuz, M. Yor, Continuous Martingales and Brownian Motion. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, 3rd edn. (Springer, Berlin, 1999)Google Scholar
  23. 23.
    V. Rivero, A law of iterated logarithm for increasing self-similar Markov processes. Stochast. Stochast. Rep. 75(6), 443–462 (2003)MATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    M. Yor, Some Aspects of Brownian Motion. Part I: Some Special Functionals. Lectures in Mathematics ETH Zürich (Birkhäuser, Basel, 1992)Google Scholar
  25. 25.
    M. Yor, Exponential Functionals of Brownian Motion and Related Processes. Springer Finance (Springer, Berlin, 2001) [With an introductory chapter by Hélyette Geman, Chapters 1, 3, 4, 8 translated from the French by Stephen S. Wilson]MATHCrossRefGoogle Scholar
  26. 26.
    M. Yor, M. Zani, Large deviations for the Bessel clock. Bernoulli 7, 351–362 (2001)MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Institut de Recherche Mathematiques de RennesRennesFrance
  2. 2.Université Versailles-Saint-Quentin, LMV UMR 8100, Bâtiment FermatVersailles-CedexFrance
  3. 3.Université d’OrléansUFR Sciences, Bâtiment de mathématiquesOrléans Cedex 2France

Personalised recommendations