An elementary derivation of moments of Hawkes processes

Abstract

Hawkes processes have been widely used in many areas, but their probability properties can be quite difficult. In this paper an elementary approach is presented to obtain moments of Hawkes processes and/or the intensity of a number of marked Hawkes processes, in which the detailed outline is given step by step; it works not only for all Markovian Hawkes processes but also for some non-Markovian Hawkes processes. The approach is simpler and more convenient than usual methods such as the Dynkin formula and martingale methods. The method is applied to one-dimensional Hawkes processes and other related processes such as Cox processes, dynamic contagion processes, inhomogeneous Poisson processes, and non-Markovian cases. Several results are obtained which may be useful in studying Hawkes processes and other counting processes. Our proposed method is an extension of the Dynkin formula, which is simple and easy to use.

Matrix Calculations for Moments of Markov Processes

“…Upon inspecting the differential equation for the n th moment of the Hawkes process intensity, one can notice that this expression depends on the moments at or below n. Thus, to compute the n th moment one must solve a system of n differential equations, meaning one must at least implicitly solve for the lower n − 1 moments first. Similarly, to solve for the (n+1) th moment of the Hawkes intensity then one must first solve for moments 1 through n, and this same pattern occurs in Cui et al [6]. Noticing this nesting pattern leads one to wonder: what other processes have moments that follow this structure?…”

“…The standard methodology for finding moments is to differentiate the moment generating function to obtain the moments, however, this is intractable for practical reasons, see for example Errais et al [13]. The problem of finding the moments of the Hawkes process is also the subject of the recent interesting research in Cui et al [6], Cui and Wu [5], works that are concurrent and independent from this one. In Cui et al [6], the authors propose a new approach for calculating moments that they construct from elementary probability arguments and also relate to the infinitesimal generator.…”

“…The problem of finding the moments of the Hawkes process is also the subject of the recent interesting research in Cui et al [6], Cui and Wu [5], works that are concurrent and independent from this one. In Cui et al [6], the authors propose a new approach for calculating moments that they construct from elementary probability arguments and also relate to the infinitesimal generator. Like the infinitesimal generator, this new methodology produces differential equations that can be solved algebraically or numerically to yield the process moments, and the authors provide closed form transient expressions up to the second moment.…”

Hawkes Processes Modeling, Inference and Control: An Overview

“…• How to properly evaluate and compare HP models among them: While there has been a lot of work regarding proposing new approaches, the comparison among existing models is often biased or incomplete. The works of [77] on how to quantify the uncertainty of the obtained models, [80] on measuring goodness-of-fit, [51] on robust identification of HPs with controlled terms, and [9] on the rigorous comparison of networked point process models are among this type of work; • Theoretical guarantees, properties and formulations of specific HP approaches, such as the works of [12] on strong mixing, [23] on the consistency of some parametric models, [15] on elementary derivations of HP momenta, and [36,37] on field master equation formulation for HPs.…”

Modeling social media contagion using Hawkes processes

“…Modelling using Hawkes have been widely used in many areas. The moments method in this context was described in [5]. Similar considerations have been performed in the context queues driven by Hawkes processes in [8], where generator method has been also applied.…”