Diffusive limits of Lipschitz functionals of Poisson measures

Abstract

Continuous time Markov Chains, Hawkes processes and many other interesting processes can be described as a 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 toward 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 diffusive limits of Markov processes and long-time behavior of Hawkes processes.