L\'evy processes conditioned to stay in a half-space with applications to directional extremes

TL;DR

This paper extends the theory of Lévy processes conditioned to stay in half-spaces to multivariate cases, linking these conditioned processes to directional extremal points and providing applications to extreme value analysis.

Contribution

It introduces a multivariate construction of Lévy processes conditioned to stay in half-spaces, generalizing previous univariate results and connecting to directional extremal behavior.

Findings

Explicit law for conditioned correlated Brownian motion as a linear transform

Limit theorem for zooming in on Lévy processes at directional infimum

Applications to analyzing extremal points in Lévy processes

Abstract

This paper provides a multivariate extension of Bertoin's pathwise construction of a L\'evy process conditioned to stay positive/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 L\'evy 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.