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

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.