Electronic Journal of Probability

Lévy processes with finite variance conditioned to avoid an interval

Leif Döring, Alexander R. Watson, and Philip Weissmann

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Abstract

Conditioning Markov processes to avoid a set is a classical problem that has been studied in many settings. In the present article we study the question if a Lévy process can be conditioned to avoid an interval and, if so, the path behavior of the conditioned process. For Lévy processes with finite second moments we show that conditioning is possible and identify the conditioned process as an $h$-transform of the original killed process. The $h$-transform is explicit in terms of successive overshoot distributions and is used to prove that the conditioned process diverges to $+\infty $ and $-\infty $ with positive probabilities.

Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 55, 32 pp.

Dates
Received: 23 July 2018
Accepted: 10 April 2019
First available in Project Euclid: 5 June 2019

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1559700305

Digital Object Identifier
doi:10.1214/19-EJP306

Zentralblatt MATH identifier
07068786

Subjects
Primary: 60J25: Continuous-time Markov processes on general state spaces 60G51: Processes with independent increments; Lévy processes 60G44: Martingales with continuous parameter

Keywords
Markov processes Lévy processes killed Lévy processes Doob $h$-transform martingales

Rights
Creative Commons Attribution 4.0 International License.

Citation

Döring, Leif; Watson, Alexander R.; Weissmann, Philip. Lévy processes with finite variance conditioned to avoid an interval. Electron. J. Probab. 24 (2019), paper no. 55, 32 pp. doi:10.1214/19-EJP306. https://projecteuclid.org/euclid.ejp/1559700305


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