Advertisement

Deep Factorisation of the Stable Process III: the View from Radial Excursion Theory and the Point of Closest Reach

  • Andreas E. KyprianouEmail author
  • Victor Rivero
  • Weerapat Satitkanitkul
Open Access
Article
  • 16 Downloads

Abstract

We compute explicitly the distribution of the point of closest reach to the origin in the path of any d-dimensional isotropic stable process, with d ≥ 2. Moreover, we develop a new radial excursion theory, from which we push the classical Blumenthal–Getoor–Ray identities for first entry/exit into a ball (cf. Blumenthal et al. Trans. Amer. Math. Soc., 99, 540–554 1961) into the more complex setting of n-tuple laws for overshoots and undershoots. We identify explicitly the stationary distribution of any d-dimensional isotropic stable process when reflected in its running radial supremum. Finally, for such processes, and as consequence of some of the analysis of the aforesaid, we provide a representation of the Wiener–Hopf factorisation of the MAP that underlies the stable process through the Lamperti–Kiu transform. Our analysis continues in the spirit of Kyprianou (Ann. Appl. Probab., 20(2), 522–564 2010) and Kyprianou et al. (2015) in that our methodology is largely based around treating stable processes as self-similar Markov processes and, accordingly, taking advantage of their Lamperti-Kiu decomposition.

Keywords

Stable processes Lévy processes Excursion theory Riesz–Bogdan–Żak transform Lamperti–Kiu transform 

Mathematics Subject Classification (2010)

Primary: 60G18 60G52 Secondary: 60G51 

Notes

Acknowledgements

The authors would like to thank Ron Doney who pointed out the distributional interpretations in Remarks 1.1 and 1.5. We would also like to thank two anonymous referees who provided two extremely helpful and thorough reports on an earlier version of this paper.

References

  1. 1.
    Alili, L., Chaumont, L., Graczyk, P., Żak, T.: Inversion, duality and doob h-transforms for self-similar Markov processes. Electron. J. Probab., 22:Paper No. 20 (2017)Google Scholar
  2. 2.
    Bertoin, J.: Lévy Processes, Volume 121 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1996)Google Scholar
  3. 3.
    Blumenthal, R.M., Getoor, R.K., Ray, D.B.: On the distribution of first hits for the symmetric stable processes. Trans. Amer. Math. Soc. 99, 540–554 (1961)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bogdan, K., Żak, T.: On Kelvin transformation. J. Theoret. Probab. 19(1), 89–120 (2006)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Caballero, M.E., Pardo, J.C., Pérez, J.L.: Explicit identities for Lévy processes associated to symmetric stable processes. Bernoulli 17(1), 34–59 (2011)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chaumont, L., Kyprianou, A.E., Pardo, J.C., Rivero, V.: Fluctuation theory and exit systems for positive self-similar Markov processes. Ann. Probab. 40(1), 245–279 (2012)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chaumont, L., Pantí, H., Rivero, V.: The Lamperti representation of real-valued self-similar Markov processes. Bernoulli 19(5B), 2494–2523 (2013)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Kuznetsov, A., Kyprianou, A.E., Pardo, J.C.: Meromorphic Lévy processes and their fluctuation identities. Ann. Appl. Probab. 22(3), 1101–1135 (2012)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kuznetsov, A., Kyprianou, A.E., Pardo, J.C., Watson, A.R.: The hitting time of zero for a stable process. Electron. J. Probab. 19(30), 26 (2014)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Kyprianou, A.E.: Fluctuations of Lévy Processes with Applications. Universitext, 2nd edn. Springer, Heidelberg (2014). Introductory lecturesCrossRefGoogle Scholar
  11. 11.
    Kyprianou, A.E.: Deep factorisation of the stable process. Electron. J. Probab., 21:Paper No. 23 (2016)Google Scholar
  12. 12.
    Kyprianou, A.E., Pardo, J.C., Rivero, V.: Exact and asymptotic n-tuple laws at first and last passage. Ann. Appl. Probab. 20(2), 522–564 (2010)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Kyprianou, A.E., Pardo, J.C., Watson, A.R.: Hitting distributions of α-stable processes via path censoring and self-similarity. Ann. Probab. 42(1), 398–430 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kyprianou, A.E., Rivero, V., Sengul, B.: Deep factorisation of the stable process II: potentials and applications. (with Victor M. Rivero and Bati Sengul). Annales de l’Instut Henri Poincaré 54(1), 343–362 (2018)CrossRefGoogle Scholar
  15. 15.
    Kyprianou, A.E., Kyprianou, A.E., Rivero, V., Satitkanitkul, W.: Conditioned real self-similar Markov processes. Stochastic Processes and their Applications 127, 1234–1254 (2017)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kyprianou, A.E.: Stable processes, self-similarity and the unit ball. arXiv:1707.04343 [math.PR] (2017)
  17. 17.
    Maisonneuve, B.: Exit systems. Ann. Probab. 3(3), 399–411 (1975)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Port, S.C., Stone, C.J.: Brownian Motion and Classical Potential Theory. Academic Press [Harcourt Brace Jovanovich Publishers], New York (1978). Probability and Mathematical StatisticszbMATHGoogle Scholar
  19. 19.
    Schilling, R.L., Song, R., Vondraček, Z.: Bernstein Functions Volume 37 of de Gruyter Studies in Mathematics, 2nd edn. Walter de Gruyter & Co., Berlin (2012). Theory and applicationsGoogle Scholar
  20. 20.
    Walsh, J.B.: Markov processes and their functionals in duality. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 24, 229–246 (1972)MathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2019

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of BathBathUK
  2. 2.CIMAT A. C.GuanajuatoMexico
  3. 3.Faculté des Sciences Bâtiment I, Département de mathématiquesUniversité d’AngersAngers cedex 01France

Personalised recommendations