The Annals of Probability

Fluctuation theory and exit systems for positive self-similar Markov processes

Loïc Chaumont, Andreas Kyprianou, Juan Carlos Pardo, and Víctor Rivero

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Abstract

For a positive self-similar Markov process, X, we construct a local time for the random set, Θ, of times where the process reaches its past supremum. Using this local time we describe an exit system for the excursions of X out of its past supremum. Next, we define and study the ladder process (R, H) associated to a positive self-similar Markov process X, namely a bivariate Markov process with a scaling property whose coordinates are the right inverse of the local time of the random set Θ and the process X sampled on the local time scale. The process (R, H) is described in terms of a ladder process linked to the Lévy process associated to X via Lamperti’s transformation. In the case where X never hits 0, and the upward ladder height process is not arithmetic and has finite mean, we prove the finite-dimensional convergence of (R, H) as the starting point of X tends to 0. Finally, we use these results to provide an alternative proof to the weak convergence of X as the starting point tends to 0. Our approach allows us to address two issues that remained open in Caballero and Chaumont [Ann. Probab. 34 (2006) 1012–1034], namely, how to remove a redundant hypothesis and how to provide a formula for the entrance law of X in the case where the underlying Lévy process oscillates.

Article information

Source
Ann. Probab., Volume 40, Number 1 (2012), 245-279.

Dates
First available in Project Euclid: 3 January 2012

Permanent link to this document
https://projecteuclid.org/euclid.aop/1325605003

Digital Object Identifier
doi:10.1214/10-AOP612

Mathematical Reviews number (MathSciNet)
MR2917773

Zentralblatt MATH identifier
1241.60019

Subjects
Primary: 60G18: Self-similar processes 60G17: Sample path properties 60J55: Local time and additive functionals

Keywords
Entrance laws exit systems excursion theory ladder processes Lamperti’s transformation Lévy processes self-similar Markov processes

Citation

Chaumont, Loïc; Kyprianou, Andreas; Pardo, Juan Carlos; Rivero, Víctor. Fluctuation theory and exit systems for positive self-similar Markov processes. Ann. Probab. 40 (2012), no. 1, 245--279. doi:10.1214/10-AOP612. https://projecteuclid.org/euclid.aop/1325605003


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