Invertibility of functionals of the Poisson process and applications

TL;DR

This paper explores conditions for invertibility of transformations on the Poisson space, introduces a new way to construct Hawkes processes, and presents a novel variational entropy representation.

Contribution

It extends invertibility results to Poisson processes, enabling new constructions of Hawkes processes and a new entropy variational formula.

Findings

Identifies entropic conditions for invertibility on Poisson space

Provides a new construction method for Hawkes processes

Establishes a new variational representation of entropy

Abstract

Following previous investigations by {\"U}st{\"u}nel [22] about the invertibility of some transformations on the Wiener space, we find some entropic conditions under which a random change of time is invertible on the Poisson space. As a consequence, we provide a new construction of Hawkes processes. We also establish a new variational representation of the entropy.