Conditioned point processes with application to Lévy bridges

Abstract

Our first result concerns a characterisation by means of a functional equation of Poisson point processes conditioned by the value of their first moment. It leads to a generalised version of Mecke's formula. En passant, it also allows to gain quantitative results about stochastic domination for Poisson point processes under linear constraints. Since bridges of a pure jump Lévy process in ${\mathbb{R}}^d$ with a height $a$ can be interpreted as a Poisson point process on space-time conditioned by pinning its first moment to $a$, our approach allows us to characterize bridges of Lévy processes by means of a functional equation. The latter result has two direct applications: first we obtain a constructive and simple way to sample Lévy bridge dynamics; second it allows to estimate the number of jumps for such bridges. We finally show that our method remains valid for linearly perturbed Lévy processes like periodic Ornstein-Uhlenbeck processes driven by Lévy noise.