Séminaire de Probabilités XLVI pp 333-343

Part of the Lecture Notes in Mathematics book series (LNM, volume 2123) | Cite as

Potentials of Stable Processes

Chapter

Abstract

For a stable process, we give an explicit formula for the potential measure of the process killed outside a bounded interval and the joint law of the overshoot, undershoot and undershoot from the maximum at exit from a bounded interval. We obtain the equivalent quantities for a stable process reflected in its infimum. The results are obtained by exploiting a simple connection with the Lamperti representation and exit problems of stable processes.

Keywords

Lévy processes Stable processes Reflected stable processes Hitting times Positive self-similar Markov processes Lamperti representation Potential measures Resolvent measures 

AMS classification (2000):

60G52 60G18 60G51 

References

  1. 1.
    E.J. Baurdoux, Some excursion calculations for reflected Lévy processes. ALEA Lat. Am. J. Probab. Math. Stat. 6, 149–162 (2009)MathSciNetMATHGoogle Scholar
  2. 2.
    J. Bertoin, Lévy Processes. Cambridge Tracts in Mathematics, vol. 121 (Cambridge University Press, Cambridge, 1996)Google Scholar
  3. 3.
    R.M. Blumenthal, R.K. Getoor, Markov Processes and Potential Theory. Pure and Applied Mathematics, vol. 29 (Academic, New York, 1968)Google Scholar
  4. 4.
    R.M. Blumenthal, R.K. Getoor, D.B. Ray, On the distribution of first hits for the symmetric stable processes. Trans. Am. Math. Soc. 99, 540–554 (1961)MathSciNetMATHGoogle Scholar
  5. 5.
    M.E. Caballero, L. Chaumont, Conditioned stable Lévy processes and the Lamperti representation. J. Appl. Probab. 43(4), 967–983 (2006). doi:10.1239/jap/1165505201CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    R.A. Doney, A.E. Kyprianou, Overshoots and undershoots of Lévy processes. Ann. Appl. Probab. 16(1), 91–106 (2006). doi:10.1214/105051605000000647CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, 7th edn. (Elsevier/Academic, Amsterdam, 2007). Translated from the Russian. Translation edited and with a preface by Alan Jeffrey and Daniel ZwillingerGoogle Scholar
  8. 8.
    A. Kuznetsov, J.C. Pardo, Fluctuations of stable processes and exponential functionals of hypergeometric Lévy processes. Acta Appl. Math. 123, 113–139 (2013). doi:10.1007/s10440-012-9718-yCrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    A.E. Kyprianou, Introductory Lectures on Fluctuations of Lévy Processes with Applications. Universitext (Springer, Berlin, 2006)MATHGoogle Scholar
  10. 10.
    A.E. Kyprianou, First passage of reflected strictly stable processes. ALEA Lat. Am. J. Probab. Math. Stat. 2, 119–123 (2006)MathSciNetMATHGoogle Scholar
  11. 11.
    A.E. Kyprianou, J.C. Pardo, V. Rivero, Exact and asymptotic n-tuple laws at first and last passage. Ann. Appl. Probab. 20(2), 522–564 (2010). doi:10.1214/09-AAP626CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    J. Lamperti, Semi-stable Markov processes. I. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 22, 205–225 (1972)CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    B.A. Rogozin, The distribution of the first hit for stable and asymptotically stable walks on an interval. Theory Probab. Appl. 17(2), 332–338 (1972). doi:10.1137/1117035CrossRefMATHGoogle Scholar
  14. 14.
    M. Sharpe, General Theory of Markov Processes. Pure and Applied Mathematics, vol. 133 (Academic, Boston, 1988)Google Scholar
  15. 15.
    J. Vuolle-Apiala, Itô excursion theory for self-similar Markov processes. Ann. Probab. 22(2), 546–565 (1994)CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.University of BathBathUK
  2. 2.University of ZürichZürichSwitzerland

Personalised recommendations