A L\'evy process on the real line seen from its supremum and max-stable processes

TL;DR

This paper derives a joint law for a process constructed from a Lévy process and its tilted version, seen from its supremum, and applies it to represent max-stable processes with a new spectral function form.

Contribution

It provides a novel expression for the joint distribution of a Lévy-based process from its supremum and introduces a new representation for max-stable processes using conditioned Lévy processes.

Findings

Derived joint law of process from its supremum and tilt

Connected max-stable processes to Lévy processes with spectral functions

Established a mixed moving maxima representation for max-stable processes

Abstract

We consider a process $Z$ on the real line composed from a L\'evy process and its exponentially tilted version killed with arbitrary rates and give an expression for the joint law of $Z$ seen from its supremum, the supremum $\overline Z$ and the time $T$ at which the supremum occurs. In fact, it is closely related to the laws of the original and the tilted L\'evy processes conditioned to stay negative and positive. The result is used to derive a new representation of stationary particle systems driven by L\'evy processes. In particular, this implies that a max-stable process arising from L\'evy processes admits a mixed moving maxima representation with spectral functions given by the conditioned L\'evy processes.