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Lévy processes conditioned to stay in a half-space with applications to directional extremes

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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.

OriginalsprogEngelsk
TidsskriftModern Stochastics: Theory and Applications
Vol/bind10
Nummer1
Sider (fra-til)59-75
Antal sider17
ISSN2351-6046
DOI
StatusUdgivet - jan. 2023

Bibliografisk note

Funding Information:
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.

Publisher Copyright:
© 2023 The Author(s).

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