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Lévy processes with finite variance conditioned to avoid an interval

Döring, L; Watson, AR; Weissmann, P; (2019) Lévy processes with finite variance conditioned to avoid an interval. Electronic Journal of Probability , 24 (55) pp. 1-32. 10.1214/19-EJP306. Green open access

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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.

Type: Article
Title: Lévy processes with finite variance conditioned to avoid an interval
Open access status: An open access version is available from UCL Discovery
DOI: 10.1214/19-EJP306
Publisher version: http://dx.doi.org/10.1214/19-EJP306
Language: English
Additional information: © The Authors. Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/).
Keywords: Markov processes, Lévy processes, killed Lévy processes, Doob h-transform, martingales
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Statistical Science
URI: https://discovery.ucl.ac.uk/id/eprint/10079241
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