Electronic Journal of Probability

On Lévy processes conditioned to stay positive.

Loïc Chaumont and Ronald Doney

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Abstract

We construct the law of Lévy processes conditioned to stay positive under general hypotheses. We obtain a Williams type path decomposition at the minimum of these processes. This result is then applied to prove the weak convergence of the law of Lévy processes conditioned to stay positive as their initial state tends to 0. We describe an absolute continuity relationship between the limit law and the measure of the excursions away from 0 of the underlying Lévy process reflected at its minimum. Then, when the Lévy process creeps upwards, we study the lower tail at 0 of the law of the height of this excursion.

Article information

Source
Electron. J. Probab., Volume 10 (2005), paper no. 28, 948-961.

Dates
Accepted: 14 July 2005
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464816829

Digital Object Identifier
doi:10.1214/EJP.v10-261

Mathematical Reviews number (MathSciNet)
MR2164035

Zentralblatt MATH identifier
1109.60039

Subjects
Primary: 60G51: Processes with independent increments; Lévy processes
Secondary: 60G17: Sample path properties

Keywords
L'evy process conditioned to stay positive path decomposition weak convergence excursion measure creeping

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

Citation

Chaumont, Loïc; Doney, Ronald. On Lévy processes conditioned to stay positive. Electron. J. Probab. 10 (2005), paper no. 28, 948--961. doi:10.1214/EJP.v10-261. https://projecteuclid.org/euclid.ejp/1464816829


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