Hostname: page-component-76fb5796d-r6qrq Total loading time: 0 Render date: 2024-04-29T23:01:53.772Z Has data issue: false hasContentIssue false
  • English
  • Français

Kalikow decomposition for counting processes with stochastic intensity and application to simulation algorithms

Part of: Special processes Stochastic processes

Published online by Cambridge University Press:  19 May 2023

Tien Cuong Phi ,
Eva Löcherbach  and
Patricia Reynaud-Bouret
Tien Cuong Phi*
Affiliation:
Université Côte d’Azur, LJAD, France
Eva Löcherbach*
Affiliation:
Université Paris 1 Panthéon-Sorbonne, France
Patricia Reynaud-Bouret*
Affiliation:
Université Côte d’Azur, CNRS, France
*
*Postal address: Université Côte d’Azur, LJAD, France. Email: cuong.tienphi@gmail.com
**Postal address: Université Paris 1 Panthéon-Sorbonne, Statistique, Analyse et Modélisation Multidisciplinaire EA 4543 et FR FP2M 2036 CNRS, France. Email: eva.locherbach@univ-paris1.fr
***Postal address: Université Côte d’Azur, CNRS, LJAD, France. Email: Patricia.Reynaud-Bouret@univ-cotedazur.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

We propose a new Kalikow decomposition for continuous-time multivariate counting processes, on potentially infinite networks. We prove the existence of such a decomposition in various cases. This decomposition allows us to derive simulation algorithms that hold either for stationary processes with potentially infinite network but bounded intensities, or for processes with unbounded intensities in a finite network and with empty past before zero. The Kalikow decomposition is not unique, and we discuss the choice of the decomposition in terms of algorithmic efficiency in certain cases. We apply these methods to several examples: the linear Hawkes process, the age-dependent Hawkes process, the exponential Hawkes process, and the Galves–Löcherbach process.

Keywords

Kalikow decompositioncounting processHawkes processperfect simulationsimulation algorithms

MSC classification

Primary: 60G55: Point processes
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 4 , December 2023 , pp. 1469 - 1500
Check for updates
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Athreya, K. B. and Ney, P. E. (1972). Branching Processes (Grundlehren der mathematischen Wissenschaften 196). Springer, New York and Heidelberg.Google Scholar
Bacry, E., Delattre, S., Hoffmann, M. and Muzy, J.-F. (2013). Some limit theorems for Hawkes processes and application to financial statistics. Stoch. Process. Appl. 123, 2475–2499.Google Scholar
Brémaud, P. (1981). Point Processes and Queues: Martingale Dynamics (Springer Series in Statistics). Springer, New York and Berlin.Google Scholar
Brémaud, P. and Massoulié, L. (1996). Stability of nonlinear Hawkes processes. Ann. Prob. 24, 1563–1588.Google Scholar
Carstensen, L., Sandelin, A., Winther, O. and Hansen, N. R. (2010). Multivariate Hawkes process models of the occurrence of regulatory elements. BMC Bioinform. 11, 1–19.CrossRefGoogle ScholarPubMed
Chen, X. (2021). Perfect sampling of Hawkes processes and queues with Hawkes arrivals. Stoch. Systems 11, 264–283.Google Scholar
Chen, X. and Wang, X. (2020). Perfect sampling of multivariate Hawkes processes. In Proceedings of the Winter Simulation Conference (WSC ’20), pp. 469--480. IEEE Press.Google Scholar
Chevallier, J. (2017). Mean-field limit of generalized Hawkes processes. Stoch. Process. Appl. 127, 3870–3912.CrossRefGoogle Scholar
Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes, Vol. I, Elementary Theory and Methods. Springer.Google Scholar
Dassios, A. and Zhao, H. (2013). Exact simulation of Hawkes process with exponentially decaying intensity. Electron. Commun. Prob. 18, 1–13.Google Scholar
Delattre, S., Fournier, N. and Hoffmann, M. (2016). Hawkes processes on large networks. Ann. Appl. Prob. 26, 216–261.Google Scholar
Ferrari, P. A. (1990). Ergodicity for spin systems with stirrings. Ann. Prob. 18, 1523–1538.CrossRefGoogle Scholar
Galves, A. and Löcherbach, E. (2013). Infinite systems of interacting chains with memory of variable length: a stochastic model for biological neural nets. J. Statist. Phys. 151, 896–921.CrossRefGoogle Scholar
Galves, A., Garcia, N., Löcherbach, E. and Orlandi, E. (2013). Kalikow-type decomposition for multicolor infinite range particle systems. Ann. Appl. Prob. 23, 1629–1659.Google Scholar
Galves, A., Löcherbach, E. and Orlandi, E. (2010). Perfect simulation of infinite range Gibbs measures and coupling with their finite range approximations. J. Statist. Phys. 138, 476–495.CrossRefGoogle Scholar
Hall, E. C. and Willett, R. M. (2016). Tracking dynamic point processes on networks. IEEE Trans. Inform. Theory 62, 4327–4346.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.Google Scholar
Hodara, P. and Löcherbach, E. (2017). Hawkes processes with variable length memory and an infinite number of components. Adv. Appl. Prob. 49, 84–107.Google Scholar
Jacod, J. (1979). Calcul Stochastique et Problèmes de Martingales (Lecture Notes Math. 714). Springer.Google Scholar
Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin, Heidelberg and New York.CrossRefGoogle Scholar
Kalikow, S. (1990). Random Markov processes and uniform martingales. Israel J. Math. 71, 33–54.CrossRefGoogle Scholar
Lewis, P. W. and Shedler, G. S. (1979). Simulation of nonhomogeneous Poisson processes by thinning. Naval Res. Logistics Quart. 26, 403–413.Google Scholar
Mascart, C., Muzy, A. and Reynaud-Bouret, P. (2020). Discrete event simulation of point processes: a computational complexity analysis on sparse graphs. Available at arXiv:2001.01702.Google Scholar
Møller, J. and Rasmussen, J. G. (2005). Perfect simulation of Hawkes processes. Adv. Appl. Prob. 37, 629–646.Google Scholar
Ogata, Y. (1981). On Lewis’ simulation method for point processes. IEEE Trans. Inform. Theory 27, 23–31.Google Scholar
Ost, G. and Reynaud-Bouret, P. (2020). Sparse space–time models: concentration inequalities and Lasso. Ann. Inst. H. Poincaré Prob. Statist. 56, 2377–2405.CrossRefGoogle Scholar
Phi, T. C., Muzy, A. and Reynaud-Bouret, P. (2020). Event-scheduling algorithms with Kalikow decomposition for simulating potentially infinite neuronal networks. SN Comput. Sci. 1, 35.Google Scholar
Raad, M. B., Ditlevsen, S. and Löcherbach, E. (2020). Stability and mean-field limits of age dependent Hawkes processes. Ann. Inst. H. Poincaré Prob. Statist. 56, 1958–1990.CrossRefGoogle Scholar
Reynaud-Bouret, P. and Roy, E. (2006). Some non asymptotic tail estimates for Hawkes processes. Bull. Belg. Math. Soc. Simon Stevin 13, 883–896.Google Scholar
Reynaud-Bouret, P. and Schbath, S. (2010). Adaptive estimation for Hawkes processes: application to genome analysis. Ann. Statist. 38, 2781–2822.Google Scholar
Scarella, G., Mascart, C., Muzy, A., Phi, T. C. and Reynaud-Bouret, P. (2021). Reconstruction de la connectivité fonctionnelle en neurosciences: une amélioration des algorithmes actuels. In 52èmes Journées de Statistique de la Société Française de Statistique (SFdS).Google Scholar