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An elementary derivation of moments of Hawkes processes

Part of: Markov processes Stochastic 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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References

Adamopoulos, L. (1975). Some counting and interval properties of the mutually-exciting processes. J. Appl. Prob. 12, 78–86.CrossRefGoogle Scholar
Brémaud, P. andMassoulie, L. (2002). Power spectra of general shot noises and Hawkes processes with a random excitation. Adv. Appl. Prob. 34, 205–222.CrossRefGoogle Scholar
Chen, J. M., Hawkes, A. G., Scalas, E. andTrinh, M. (2018). Performance of information criteria for selection of Hawkes process models of financial data. Quant. Finance 18 (2), 225–235.CrossRefGoogle Scholar
Cox, D. R. (1955). Some statistical methods connected with series of events. J. R. Statist. Soc. B 17, 129–164.Google Scholar
Cui, L. R., Chen, Z. L. andGao, H. D. (2018). Reliability for systems with self-healing effect under shock models. Qual. Technol. Quant. Manag. 15, 551–567.CrossRefGoogle Scholar
Cui, L. R., Li, Z. P. andYi, H. (2019). Partial self-exciting point processes and their parameter estimations. Commun. Statist. Simul. Comput. 48 (10), 2913–2935.CrossRefGoogle Scholar
Dassios, A. andJang, J. W. (2003). Pricing of catastrophe reinsurance and derivatives using the Cox process with shot noise intensity. Finance Stoch. 7, 73–95.CrossRefGoogle Scholar
Dassios, A. andZhao, H. B. (2011). A dynamic contagion process. Adv. Appl. Prob. 43, 814–846.CrossRefGoogle Scholar
Davis, M. H. A. (1984). Piecewise-deterministic Markov processes: a general class of nondiffusion stochastic models. J. R. Statist. Soc. B 46, 353–388.Google Scholar
Daw, A. andPender, J. (2018). Queues driven by Hawkes processes. Stoch. Sys. 12, 192–229.CrossRefGoogle Scholar
Daw, A. andPender, J. (2019). Matrix calculations for moments of Markov processes. Working paper, by personal communication. Available from .Google Scholar
Duffie, D., Filipovic, D. andSchachermayer, W. (2003). Affine processes and applications in finance. Ann. Appl. Prob. 13, 984–1053.Google Scholar
Errais, E., Giesecke, K. andGoldberg, L. R. (2010). Affine point processes and portfolio credit risk. SIAM J. Financial Math. 1, 642–665.CrossRefGoogle Scholar
Hawkes, A. G. (1971). Spectra of some self-exciting and mutually exciting point processes. Biometrika 58, 83–90.CrossRefGoogle Scholar
Hawkes, A. G. (1971). Point spectra of some mutually exciting point processes. J. R. Statist. Soc. B 33 (3), 438–443.Google Scholar
Hawkes, A. G. (1972). Spectra of some mutually exciting point processes with associated variables. In Stochastic Point Processes, ed. P. A. W. Lewis, pp. 261–271. Wiley, New York.Google Scholar
Hawkes, A. G. andOakes, D. (1974). A cluster representation of a self-exciting process. J. Appl. Prob. 11, 493–503.CrossRefGoogle Scholar
Li, Z. P., Cui, L. R. andChen, J. H. (2018). Traffic accident modelling via self-exciting point processes. Reliab. Eng. Syst. Safe. 180, 312–320.CrossRefGoogle Scholar
Mohler, G., Short, M., Brantingham, P., Schoenberg, F. andTita, G. (2011). Self-exciting point process modeling of crime. J. Amer. Statist. Assoc. 106 (493), 100–108.CrossRefGoogle Scholar
Møller, J. (2003). Shot noise Cox processes. Adv. Appl. Prob. 35, 614–640.CrossRefGoogle Scholar
Oakes, D. (1975). The Markovian self-exciting process. J. Appl. Prob. 12, 69–77.CrossRefGoogle Scholar
Ogata, Y. (1988). Statistical models for earthquake occurrences and residual analysis for point processes. J. Amer. Statist. Assoc. 83 (401), 9–27.CrossRefGoogle Scholar
Veen, A. andSchoenberg, F. P. (2008). Estimation of space-time branching process models in seismology using an EM-type algorithm. J. Amer. Statist. Assoc. 103, 614–624.CrossRefGoogle Scholar
Zhu, L. (2013). Nonlinear Hawkes processes. Doctoral thesis, New York University.Google Scholar