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Lévy processes conditioned to stay in a half-space with applications to directional extremes
Volume 10, Issue 1 (2023), pp. 59–75
Jevgenijs Ivanovs ORCID icon link to view author Jevgenijs Ivanovs details   Jakob D. Thøstesen ORCID icon link to view author Jakob D. Thøstesen details  

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https://doi.org/10.15559/22-VMSTA217
Pub. online: 25 November 2022      Type: Research Article      Open accessOpen Access

Received
24 July 2022
Accepted
11 November 2022
Published
25 November 2022

Abstract

This paper provides a multivariate extension of Bertoin’s pathwise construction of a Lévy process conditioned to stay positive or negative. Thus obtained processes conditioned to stay in half-spaces are closely related to the original process on a compact time interval seen from its directional extremal points. In the case of a correlated Brownian motion the law of the conditioned process is obtained by a linear transformation of a standard Brownian motion and an independent Bessel-3 process. Further motivation is provided by a limit theorem corresponding to zooming in on a Lévy process with a Brownian part at the point of its directional infimum. Applications to zooming in at the point furthest from the origin are envisaged.

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© 2023 The Author(s). Published by VTeX
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Keywords
Conditioning to stay positive directional extremes exchangeability local behavior Sparre Andersen identity

MSC2010
60G51 60G17 60F17

Funding
The authors gratefully acknowledge the financial support of Sapere Aude Starting Grant 8049-00021B “Distributional Robustness in Assessment of Extreme Risk” from Independent Research Fund Denmark.

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