Conditioned point processes with application to L\'evy bridges

TL;DR

This paper characterizes conditioned Poisson point processes and Lévy bridges using functional equations, enabling new sampling methods and jump estimates for Lévy processes and their perturbations.

Contribution

It introduces a functional equation characterization of Lévy bridges as conditioned Poisson point processes, extending Mecke's formula and providing practical sampling and analysis tools.

Findings

Characterization of Lévy bridges via functional equations

A simple method for sampling Lévy bridge dynamics

Estimates on the number of jumps in 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 L\'evy 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 L\'evy processes by means of a functional equation. The latter result has two direct applications: first we obtain a constructive and simple way to sample L\'evy 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 L\'evy processes like periodic Ornstein-Uhlenbeck processes driven by L\'evy noise.