Exponential moments for Hawkes processes under minimal assumptions

TL;DR

This paper establishes that stationary linear Hawkes processes have exponential moments for the count of points in any bounded region, under minimal assumptions, using a mass transport principle argument.

Contribution

It proves exponential moment bounds for Hawkes processes with minimal assumptions, extending previous results and providing explicit dependence on process parameters.

Findings

Exponential moments exist for counts in bounded regions.

Bounds depend explicitly on base rates and interaction matrix.

Results apply under minimal spectral radius conditions.

Abstract

We prove that the number of points of a stationary linear Hawkes process lying in any bounded subset of the real line has exponential moments, without any other assumption than the one needed for existence of such stationary process, namely the spectral radius of the matrix of L1 norms of interaction functions is smaller than one. The proof relies on a mass transport principle argument. We also specify the dependence of the bounds with respect to the base rates and the matrix of L1 norms of interaction functions defining the Hawkes process and give a functional version of the result.