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Perfect simulation of Hawkes processes

Part of: Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Jesper Møller  and
Jakob G. Rasmussen
Jesper Møller*
Affiliation:
Aalborg University
Jakob G. Rasmussen*
Affiliation:
Aalborg University
*
∗ Postal address: Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7G, DK-9220 Aalborg, Denmark.
∗ Postal address: Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7G, DK-9220 Aalborg, Denmark.
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Abstract

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Our objective is to construct a perfect simulation algorithm for unmarked and marked Hawkes processes. The usual straightforward simulation algorithm suffers from edge effects, whereas our perfect simulation algorithm does not. By viewing Hawkes processes as Poisson cluster processes and using their branching and conditional independence structures, useful approximations of the distribution function for the length of a cluster are derived. This is used to construct upper and lower processes for the perfect simulation algorithm. A tail-lightness condition turns out to be of importance for the applicability of the perfect simulation algorithm. Examples of applications and empirical results are presented.

Keywords

Approximate simulationdominated coupling from the pastedge effectexact simulationmarked Hawkes processmarked point processperfect simulationpoint processPoisson cluster processthinningupper processlower process

MSC classification

Primary: 60G55: Point processes
Secondary: 68U20: Simulation
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 37 , Issue 3 , September 2005 , pp. 629 - 646
Copyright
Copyright © Applied Probability Trust 2005 

References

Apostol, T. M. (1974). Mathematical Analysis. Addison-Wesley, Reading, MA.Google Scholar
Asmussen, S. (1987). Applied Probability and Queues. John Wiley, Chichester.Google Scholar
Brémaud, P. and Massoulié, L. (1996). Stability of nonlinear Hawkes processes. Ann. Prob. 24, 1563–1588.CrossRefGoogle Scholar
Brémaud, P. and Massoulié, L. (2001). Hawkes branching point processes without ancestors. J. Appl. Prob. 38, 122–135.CrossRefGoogle Scholar
Brémaud, P., Nappo, G. and Torrisi, G. (2002). Rate of convergence to equilibrium of marked Hawkes processes. J. Appl. Prob. 39, 123–136.CrossRefGoogle Scholar
Brix, A. and Kendall, W. S. (2002). Simulation of cluster point processes without edge effects. Adv. Appl. Prob. 34, 267–280.CrossRefGoogle Scholar
Chornoboy, E. S., Schramm, L. P. and Karr, A. F. (1988). Maximum likelihood identification of neural point process systems. Biol. Cybernet. 59, 265–275.CrossRefGoogle ScholarPubMed
Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes, Vol. 1, Elementary Theory and Methods, 2nd edn. Springer, New York.Google Scholar
Dwass, M. (1969). The total progeny in a branching process and a related random walk. J. Appl. Prob. 6, 682–686.CrossRefGoogle Scholar
Hawkes, A. G. (1971). Point spectra of some mutually exciting point processes. J. R. Statist. Soc. B 33, 438–443.Google Scholar
Hawkes, A. G. (1971). Spectra of some self-exciting and mutually exciting point processes. Biometrika 58, 83–90.CrossRefGoogle Scholar
Hawkes, A. G. (1972). Spectra of some mutually exciting point processes with associated variables. In Stochastic Point Processes, ed. Lewis, P. A. W., John Wiley, New York, pp. 261–271.Google Scholar
Hawkes, A. G. and Adamopoulos, L. (1973). Cluster models for earthquakes – regional comparisons. Bull. Internat. Statist. Inst. 45, 454–461.Google Scholar
Hawkes, A. G. and Oakes, D. (1974). A cluster representation of a self-exciting process. J. Appl. Prob. 11, 493–503.CrossRefGoogle Scholar
Jagers, P. (1975). Branching Processes with Biological Applications. John Wiley, London.Google Scholar
Kendall, W. S. and Møller, J. (2000). Perfect simulation using dominating processes on ordered spaces, with application to locally stable point processes. Adv. Appl. Prob. 32, 844–865.CrossRefGoogle Scholar
Møller, J. (2003). Shot noise Cox processes. Adv. Appl. Prob. 35, 614–640.CrossRefGoogle Scholar
Møller, J. and Rasmussen, J. G. (2005). Approximate simulation of Hawkes processes. Submitted.Google Scholar
Møller, J. and Torrisi, G. L. (2005). Generalised shot noise Cox processes. Adv. Appl. Prob. 37, 48–74.CrossRefGoogle Scholar
Møller, J. and Torrisi, G. L. (2005). Perfect and approximate simulation of spatial Hawkes processes. In preparation.Google Scholar
Møller, J. and Waagepetersen, R. P. (2004). Statistical Inference and Simulation for Spatial Point Processes. Chapman and Hall, Boca Raton, FL.Google Scholar
Ogata, Y. (1988). Statistical models for earthquake occurrences and residual analysis for point processes. J. Amer. Statist. Assoc. 83, 9–27.CrossRefGoogle Scholar
Ogata, Y. (1998). Space-time point-process models for earthquake occurrences. Ann. Inst. Statist. Math. 50, 379–402.CrossRefGoogle Scholar
Propp, J. G. and Wilson, D. B. (1996). Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures Algorithms 9, 223–252.3.0.CO;2-O>CrossRefGoogle Scholar
Ripley, B. D. (1987). Stochastic Simulation. John Wiley, New York.CrossRefGoogle Scholar
Rudin, W. (1987). Real and Complex Analysis. McGraw-Hill, New York.Google Scholar
Vere-Jones, D. and Ozaki, T. (1982). Some examples of statistical inference applied to earthquake data. Ann. Inst. Statist. Math. 34, 189–207.CrossRefGoogle Scholar