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An elementary derivation of moments of Hawkes processes

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  29 April 2020

Lirong Cui ,
Alan Hawkes  and
He Yi
Lirong Cui*
Affiliation:
Beijing Institute of Technology
Alan Hawkes*
Affiliation:
Swansea University
He Yi*
Affiliation:
Beijing Institute of Technology
*
*Postal address: School of Management & Economics, Beijing Institute of Technology, Beijing 100081, China.
***Postal address: School of Management, Swansea University, Fabian Way, Swansea SA1 8EN, UK.
****Postal address: School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China.
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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.

Keywords

Hawkes processDynkin formulamomentscounting processgamma decay function

MSC classification

Primary: 60G55: Point processes
Secondary: 60G57: Random measures 60G44: Martingales with continuous parameter 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Original Article
Information
Advances in Applied Probability , Volume 52 , Issue 1 , March 2020 , pp. 102 - 137
Copyright
© Applied Probability Trust 2020

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