Lévy processes conditioned to stay in a half-space with applications to directional extremes
Volume 10, Issue 1 (2023), pp. 59–75
Pub. online: 25 November 2022
Type: Research Article
Open Access
Open Access
Received
24 July 2022
24 July 2022
Accepted
11 November 2022
11 November 2022
Published
25 November 2022
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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