Electronic Communications in Probability

Right inverses of Levy processes: the excursion measure in the general case

Mladen Savov and Matthias Winkel

Full-text: Open access

Abstract

This article is about right inverses of Lévy processes as first introduced by Evans in the symmetric case and later studied systematically by the present authors and their co-authors. Here we add to the existing fluctuation theory an explicit description of the excursion measure away from the (minimal) right inverse. This description unifies known formulas in the case of a positive Gaussian coefficient and in the bounded variation case. While these known formulas relate to excursions away from a point starting negative continuously, and excursions started by a jump, the present description is in terms of excursions away from the supremum continued up to a return time. In the unbounded variation case with zero Gaussian coefficient previously excluded, excursions start negative continuously, but the excursion measures away from the right inverse and away from a point are mutually singular. We also provide a new construction and a new formula for the Laplace exponent of the minimal right inverse.

Article information

Source
Electron. Commun. Probab., Volume 15 (2010), paper no. 51, 572-584.

Dates
Accepted: 12 December 2010
First available in Project Euclid: 6 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465243994

Digital Object Identifier
doi:10.1214/ECP.v15-1590

Mathematical Reviews number (MathSciNet)
MR2746335

Zentralblatt MATH identifier
1226.60071

Subjects
Primary: 60G51: Processes with independent increments; Lévy processes

Keywords
Levy process right inverse subordinator fluctuation theory excursion

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Savov, Mladen; Winkel, Matthias. Right inverses of Levy processes: the excursion measure in the general case. Electron. Commun. Probab. 15 (2010), paper no. 51, 572--584. doi:10.1214/ECP.v15-1590. https://projecteuclid.org/euclid.ecp/1465243994


Export citation

References

  • J. Bertoin. Lévy Processes. Cambridge University Press 1996.
  • R. Doney. Fluctuation theory for Lévy processes. Lectures from the 35th Summer School on Probability Theory held in Saint-Flour. Lecture Notes in Mathematics 1897. Springer Berlin 2007.
  • R.A. Doney, A.E. Kyprianou. Overshoots and undershoots of Lévy processes. Ann. Appl. Prob. 16 (2006), No. 1, 91-106.
  • R. Doney, M. Savov. Right inverses of Lévy processes. Ann. Prob. 38 (2010), No. 4, 1390-1400.
  • S. Evans. Right inverses of non-symmetric Lévy processes and stationary stopped local times. Probab. Theory Related Fields. 118 (2000), 37-48.
  • K. Sato. Lévy processes and infinitely divisible distributions. Cambridge University Press 1999.
  • T. Simon. Subordination in the wide sense for Lévy processes. Probab. Theory Related Fields. 115 (1999), 445-477.
  • M. Winkel. Right inverses of non-symmetric Lévy processes. Ann. Prob. 30 (2002), No. 1, 382-415.