The Malliavin-Stein method for Hawkes functionals

TL;DR

This paper develops a new method combining Malliavin calculus and Stein's method to quantify the convergence of Hawkes process functionals to Gaussian distributions, providing the first Berry-Esséen bounds for such processes.

Contribution

It introduces a Malliavin calculus framework for compound Hawkes processes and derives novel Berry-Esséen bounds for their CLTs, filling a gap in the literature.

Findings

Established bounds on Wasserstein distance for Hawkes functionals

Provided the first Berry-Esséen bounds for compound Hawkes processes

Demonstrated the effectiveness of the combined Malliavin-Stein approach

Abstract

In this paper, following Nourdin-Peccati's methodology, we combine the Malliavin calculus and Stein's method to provide general bounds on the Wasserstein distance between functionals of a compound Hawkes process and a given Gaussian density. To achieve this, we rely on the Poisson embedding representation of an Hawkes process to provide a Malliavin calculus for the Hawkes processes, and more generally for compound Hawkes processes. As an application, we close a gap in the literature by providing the first Berry-Ess\'een bounds associated to Central Limit Theorems for the compound Hawkes process.