You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:


Log in to your personal account or through your institution.

Perfect Simulation of Hawkes Processes

Jesper Møller and Jakob G. Rasmussen
Advances in Applied Probability
Vol. 37, No. 3 (Sep., 2005), pp. 629-646
Stable URL:
Page Count: 18
Subjects: Statistics
  • Download PDF
  • Add to My Lists
  • Cite this Item
Perfect Simulation of Hawkes Processes
We're having trouble loading this content. Download PDF instead.


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.

Notes and References

This item contains 27 references.

  • 1
    APOSTOL, T. M. (1974). Mathematical Analysis. Addison-Wesley, Reading, MA.
  • 2
    ASMUSSEN, S. (1987). Applied Probability and Queues. John Wiley, Chichester.
  • 3
    BRÉMAUD, P. AND MASSOULIÉ, L. (1996). Stability of nonlinear Hawkes processes. Ann. Prob. 24, 1563-1588.
  • 4
    BRÉMAUD, P. AND MASSOULIÉ, L. (2001). Hawkes branching point processes without ancestors. J. Appl. Prob. 38, 122-135.
  • 5
    BRÉMAUD, P., NAPPO, G. AND TORRISI, G. (2002). Rate of convergence to equilibrium of marked Hawkes processes. J. Appl. Prob. 39, 123-136.
  • 6
    BRIX, A. AND KENDALL, W. S. (2002). Simulation of cluster point processes without edge effects. Adv. Appl. Prob. 34, 267-280.
  • 7
    CHORNOBOY, E. S., SCHRAMM, L. P. AND KARR, A. F. (1988). Maximum likelihood identification of neural point process systems. Biol. Cybernet. 59, 265-275.
  • 8
    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.
  • 9
    DWASS, M. (1969). The total progeny in a branching process and a related random walk. J. Appl. Prob. 6, 682-686.
  • 10
    HAWKES, A. G. (1971). Point spectra of some mutually exciting point processes. J. R. Statist. Soc. B 33, 438-443.
  • 11
    HAWKES, A. G. (1971). Spectra of some self-exciting and mutually exciting point processes. Biometrika 58, 83-90.
  • 12
    HAWKES, A. G. (1972). Spectra of some mutually exciting point processes with associated variables. In Stochastic Point Processes, ed. P. A. W. Lewis, John Wiley, New York, pp. 261-271.
  • 13
    HAWKES, A. G. AND ADAMOPOULOS, L. (1973). Cluster models for earthquakes - regional comparisons. Bull. Internat. Statist. Inst. 45, 454-461.
  • 14
    HAWKES, A. G. AND OAKES, D. (1974). A cluster representation of a self-exciting process. J. Appl. Prob. 11, 493-503.
  • 15
    JAGERS, P. (1975). Branching Processes with Biological Applications. John Wiley, London.
  • 16
    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.
  • 17
    MØLLER, J. (2003). Shot noise Cox processes. Adv. Appl. Prob. 35, 614-640.
  • 18
    MØLLER, J. AND RASMUSSEN, J. G. (2005). Approximate simulation of Hawkes processes. Submitted.
  • 19
    MØLLER, J. AND TORRISI, G. L. (2005). Generalised shot noise Cox processes. Adv. Appl. Prob. 37, 48-74.
  • 20
    MØLLER, J. AND TORRISI, G. L. (2005). Perfect and approximate simulation of spatial Hawkes processes. In preparation.
  • 21
    MØLLER, J. AND WAAGEPETERSEN, R. P. (2004). Statistical Inference and Simulationfor Spatial Point Processes. Chapman and Hall, Boca Raton, FL.
  • 22
    OGATA, Y. (1988). Statistical models for earthquake occurrences and residual analysis for point processes. J. Amer Statist. Assoc. 83, 9-27.
  • 23
    OGATA, Y. (1998). Space-time point-process models for earthquake occurrences. Ann. Inst. Statist. Math. 50, 379-402.
  • 24
    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.
  • 25
    RIPLEY, B. D. (1987). Stochastic Simulation. John Wiley, New York.
  • 26
    RUDIN, W. (1987). Real and Complex Analysis. McGraw-Hill, New York.
  • 27
    VERE-JONES, D. AND OZAKI, T. (1982). Some examples of statistical inference applied to earthquake data. Ann. Inst. Statist. Math. 34, 189-207.