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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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

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

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

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Zentralblatt MATH identifier

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

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

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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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