Exponential moments for Hawkes processes under minimal assumptions
type: Item
instance of: scholarly article
title: Exponential moments for Hawkes processes under minimal assumptions
author: Théo Leblanc
published in: Electronic Communications in Probability
publication date: 2024-12-20T00:00:00Z
review text: In this article the author proves that a stationary Hawkes process has exponential moments under the minimal spectral radius condition given by \textit{P. Brémaud} and \textit{L. Massoulié} [Ann. Probab. 24, No. 3, 1563--1588 (1996; Zbl 0870.60043)],\N\[\N\operatorname{SpR} (\|h^{m}_{m^{\prime}}\|_{1})_{m,m^{\prime} \in M}<1 \N\]\Nwhere \(h^{m}_{m^{\prime}}\) is the Lebesgue integrable interaction function. Under the spectral radius condition, a linear stationary Hawkes process \(N = (N^{1},N^{2},\dots,N^{M})\) with intensity\N\[\N\lambda^{m}_{t} = \mu_{m} + \sum_{m^{\prime} \in M} \int_{-\infty}^{t-} h_{m^{\prime}}^{m}(t-s)dN_{s}^{m^{\prime}}\N\]\Nexists and has exponential moments. The author also considers a cluster representation such as the Galton-Watson tree and gives a functional version bound.
reviewed by: Haoming Wang
Mathematics Subject Classification ID: 60G55
Mathematics Subject Classification ID: 60J80
zbMATH DE Number: 7960010
zbMATH Keywords: Hawkes processes
zbMATH Keywords: cluster representation
zbMATH Keywords: exponential moments
MaRDI profile type: Publication
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DOI: 10.1214/24-ECP625
description: scientific article; zbMATH DE number 7960010