Malliavin-Stein method for the multivariate compound Hawkes process

TL;DR

This paper develops bounds on how quickly functionals of multivariate compound Hawkes processes converge to Gaussian vectors, using Malliavin calculus, and applies this to CLTs with exponential kernels as observation time increases.

Contribution

It introduces a Malliavin-Stein approach to quantify convergence rates of multivariate compound Hawkes processes to Gaussian limits.

Findings

Derived upper bounds on the d2 distance to Gaussian vectors.

Established convergence rates in CLTs for Hawkes processes with exponential kernels.

Provided a framework for analyzing functionals of Hawkes processes using Malliavin calculus.

Abstract

In this paper, we provide upper bounds on the d2 distance between a large class of functionals of a multivariate compound Hawkes process and a given Gaussian vector. This is proven using Malliavin's calculus defined on an underlying Poisson embedding. The upper bound is then used to infer the speed of convergence of Central Limit Theorems for the multivariate compound Hawkes process with exponential kernels as the observation time T goes to infinity.