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.
Part of this work was done while the second author was at the University of Bath, UK, and at CIMAT, Mexico.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
E.J. Baurdoux, Some excursion calculations for reflected Lévy processes. ALEA Lat. Am. J. Probab. Math. Stat. 6, 149–162 (2009)
J. Bertoin, Lévy Processes. Cambridge Tracts in Mathematics, vol. 121 (Cambridge University Press, Cambridge, 1996)
R.M. Blumenthal, R.K. Getoor, Markov Processes and Potential Theory. Pure and Applied Mathematics, vol. 29 (Academic, New York, 1968)
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)
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/1165505201
R.A. Doney, A.E. Kyprianou, Overshoots and undershoots of Lévy processes. Ann. Appl. Probab. 16(1), 91–106 (2006). doi:10.1214/105051605000000647
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 Zwillinger
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-y
A.E. Kyprianou, Introductory Lectures on Fluctuations of Lévy Processes with Applications. Universitext (Springer, Berlin, 2006)
A.E. Kyprianou, First passage of reflected strictly stable processes. ALEA Lat. Am. J. Probab. Math. Stat. 2, 119–123 (2006)
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-AAP626
J. Lamperti, Semi-stable Markov processes. I. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 22, 205–225 (1972)
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:10.1137/1117035
M. Sharpe, General Theory of Markov Processes. Pure and Applied Mathematics, vol. 133 (Academic, Boston, 1988)
J. Vuolle-Apiala, Itô excursion theory for self-similar Markov processes. Ann. Probab. 22(2), 546–565 (1994)
Acknowledgements
We would like to thank the referee for his careful reading of this paper.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Kyprianou, A.E., Watson, A.R. (2014). Potentials of Stable Processes. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités XLVI. Lecture Notes in Mathematics(), vol 2123. Springer, Cham. https://doi.org/10.1007/978-3-319-11970-0_12
Download citation
DOI: https://doi.org/10.1007/978-3-319-11970-0_12
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-11969-4
Online ISBN: 978-3-319-11970-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)