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Journal Articles The Annals of Applied Probability Year : 2024

Diffusive limits of Lipschitz functionals of Poisson measures

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.
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Dates and versions

hal-03283778 , version 1 (12-07-2021)
hal-03283778 , version 2 (15-12-2022)

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Eustache Besançon, Laure Coutin, Laurent Decreusefond, Pascal Moyal. Diffusive limits of Lipschitz functionals of Poisson measures. The Annals of Applied Probability, 2024, 34 (1A), pp.555-584. ⟨10.1214/23-AAP1972⟩. ⟨hal-03283778v2⟩
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