Advertisement

Probability Theory and Related Fields

, Volume 129, Issue 2, pp 261–289 | Cite as

Large deviations for squares of Bessel and Ornstein-Uhlenbeck processes

  • C. Donati-Martin
  • A. Rouault
  • M. Yor
  • M. Zani
Article

Abstract.

Let (X t (δ) ,t≥0) be the BESQδ process starting at δx. We are interested in large deviations as \({{\delta \rightarrow \infty}}\) for the family {δ−1 X t (δ) ,tT}δ, – or, more generally, for the family of squared radial OUδ process. The main properties of this family allow us to develop three different approaches: an exponential martingale method, a Cramér–type theorem, thanks to a remarkable additivity property, and a Wentzell–Freidlin method, with the help of McKean results on the controlled equation. We also derive large deviations for Bessel bridges.

Keywords

Bessel processes Ornstein-Uhlenbeck processes Additivity property Large deviation principle 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Azencott, R.: Grandes déviations et applications. In: Ecole d’été de Probabilités de Saint-Flour VIII, Lecture Notes in Mathematics, 774, Springer-Verlag, 1978, pp. 1–176Google Scholar
  2. 2.
    Barlow, M.T., Yor, M.: Semi-martingale inequalities via the Garsia-Rodemich-Rumsey lemma and applications to local time. J. Funct. Anal. 49, 198–299 (1982)Google Scholar
  3. 3.
    BenArous, G., Ledoux, M.: Grandes Déviations de Freidlin-Wentzell en Norme Höldérienne. In: Séminaire de Probabilités XXVIII, Lecture Notes in Mathematics, 1583, Springer-Verlag, 1994, pp. 293–299Google Scholar
  4. 4.
    Biane, P., Yor, M.: Variations sur une formule de Paul Lévy. Ann. Inst. H. Poincaré Probab. Statist. 23, 359–377 (1987)Google Scholar
  5. 5.
    Dawson, D.A., Gartner, J.: Large deviations for Mc-Kean Vlasov limit of weakly interacting diffusions. Stochastics 20, 247–308 (1987)Google Scholar
  6. 6.
    Dembo, A., Zeitouni, O.: Large deviations techniques and applications. Second edition, Springer, 1998Google Scholar
  7. 7.
    Deuschel, J.D., Stroock, D.W.: Large deviations. Academic Press, 1989Google Scholar
  8. 8.
    Feng, S.: The behaviour near the boundary of some degenerate diffusions under random perturbation. Stochastic models (Ottawa ON 1998), Providence RI, Amer. Math. Soc., 2000, pp. 115–123Google Scholar
  9. 9.
    Fernique, X.: Régularité des trajectoires des fonctions aléatoires gaussiennes. In: Ecole d’été de Probabilités de Saint-Flour IVLecture Notes in Mathematics, 480 Springer-Verlag, 1974, pp. 1–97Google Scholar
  10. 10.
    Freidlin, M.I., Wentzell, A.D.: Random perturbations of dynamical systems. Springer-Verlag, 1984Google Scholar
  11. 11.
    Lamperti, J.: Continuous state branching processes. Bull. Am. Math. Soc. 73, 382–386 (1967)Google Scholar
  12. 12.
    Lévy, P.: Wiener’s random function and other Laplacian random functions. In: Second Berkeley Symposium, Berkeley and Los Angeles, University of California Press, 1951, pp. 171–187Google Scholar
  13. 13.
    Liptser, R.S., Pukhalskii, A.A.: Limit Theorems on Large Deviations for Semimartingales. Stochastics Stochastics Rep. 38, 201–249 (1992)Google Scholar
  14. 14.
    McKean, H.P.: The Bessel motion and a singular integral equation. Mem. Coll. Sci. Univ. Kyoto. Ser. A Math. 33, 317–322 (1960/1961)Google Scholar
  15. 15.
    McKean H.P.: Stochastic integrals. Academic Press, New York, 1969Google Scholar
  16. 16.
    Pitman, J.: Cyclically stationary Brownian local time processes. Probab. Theory Related. Fields 106, 299–329 (1996)Google Scholar
  17. 17.
    Pitman, J., Yor, M.: A decomposition of Bessel Bridges. Z. Wahrscheinlichkeitstheorie. Verw. Geb. 59, 425–457 (1982)Google Scholar
  18. 18.
    Revuz, D., Yor, M.: Continuous martingales and Brownian motion. 3rd ed., Springer, Berlin, 1999Google Scholar
  19. 19.
    Shiga, T., Watanabe, S.: Bessel diffusions as a one–parameter family of diffusion. processes Z. Wahrscheinlichkeitstheorie. Verw. Geb. 27, 37–46 (1973)Google Scholar
  20. 20.
    Stroock, D.W., Varadhan, S.R.S.: Multidimensional diffusion processes. Springer-Verlag, Berlin, 1979Google Scholar
  21. 21.
    Wentzell, A., Freidlin, M.: On small random perturbations of dynamical systems. Russian Math. Surveys 25, 1–55 (1970)Google Scholar
  22. 22.
    Yor, M.: On some exponential functionals of Brownian motion. Adv. Appl. Probab. 24, 509–531 (1992)Google Scholar
  23. 23.
    Yor, M.: Some aspects of Brownian motion Part I: Some special functionals. In: Lectures in Mathematics, ETH Zürich, Birkhäuser, Basel, 1992Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Laboratoire de Probabilités et Modèles AléatoiresParisFrance
  2. 2.LAMA, Bâtiment FermatUniversité de VersaillesVersaillesFrance
  3. 3.Laboratoire de Probabilités et Modèles AléatoiresParisFrance
  4. 4.Laboratoire d’Analyse et de Mathématiques AppliquéesCréteilFrance

Personalised recommendations