Diffusive limits of Lipschitz functionals of Poisson measures

TL;DR

This paper extends convergence results of Poisson measures to diffusive limits for Markov processes and Hawkes processes using Lipschitz continuity of operations, building on Stein's method.

Contribution

It demonstrates that operations like time change and convolution preserve Lipschitz continuity, enabling broader convergence results for stochastic processes driven by Poisson measures.

Findings

Extended convergence rates to diffusive limits for Markov processes.

Applied Lipschitz continuity to operations on Poisson-driven processes.

Provided a framework for analyzing long-time behavior of Hawkes processes.

Abstract

Continuous Time Markov Chains, Hawkes processes and many other interesting processes can be described as solution of stochastic differential equations driven by Poisson measures. Previous works, using the Stein's method, give the convergence rate of a sequence of renormalized Poisson measures towards the Brownian motion in several distances, constructed on the model of the Kantorovitch-Rubinstein (or Wasserstein-1) distance. We show that many operations (like time change, convolution) on continuous functions are Lipschitz continuous to extend these quantified convergences to diffuse limits of Markov processes and long-time behavior of Hawkes processes.