Potentials of Stable Processes
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.
KeywordsLévy 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
We would like to thank the referee for his careful reading of this paper.
- 2.J. Bertoin, Lévy Processes. Cambridge Tracts in Mathematics, vol. 121 (Cambridge University Press, Cambridge, 1996)Google Scholar
- 3.R.M. Blumenthal, R.K. Getoor, Markov Processes and Potential Theory. Pure and Applied Mathematics, vol. 29 (Academic, New York, 1968)Google Scholar
- 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
- 14.M. Sharpe, General Theory of Markov Processes. Pure and Applied Mathematics, vol. 133 (Academic, Boston, 1988)Google Scholar