Process-level large deviations for nonlinear Hawkes point processes
Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 3, p. 845-871

In this paper, we prove a process-level, also known as level-3 large deviation principle for a very general class of simple point processes, i.e. nonlinear Hawkes process, with a rate function given by the process-level entropy, which has an explicit formula.

Dans cet article nous prouvons un principe de grandes déviations de niveau trois pour une classe très générale de processus ponctuels, c'est à dire les processus de Hawkes non-linéaires ; nous obtenons une formule explicite pour la fonctionnelle de taux, donnée par l'entropie au niveau du processus.

DOI : https://doi.org/10.1214/12-AIHP532
Classification:  60G55,  60F10
Keywords: large deviations, rare events, point processes, Hawkes processes, self-exciting processes
@article{AIHPB_2014__50_3_845_0,
     author = {Zhu, Lingjiong},
     title = {Process-level large deviations for nonlinear Hawkes point processes},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {50},
     number = {3},
     year = {2014},
     pages = {845-871},
     doi = {10.1214/12-AIHP532},
     zbl = {1296.60129},
     mrnumber = {3224291},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2014__50_3_845_0}
}
Zhu, Lingjiong. Process-level large deviations for nonlinear Hawkes point processes. Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 3, pp. 845-871. doi : 10.1214/12-AIHP532. http://www.numdam.org/item/AIHPB_2014__50_3_845_0/

[1] E. Bacry, S. Delattre, M. Hoffmann and J. F. Muzy. Scaling limits for Hawkes processes and application to financial statistics. Preprint, 2012. Available at arXiv:1202.0842. | Zbl 1292.60032

[2] C. Bordenave and G. L. Torrisi. Large deviations of Poisson cluster processes. Stoch. Models 23 (2007) 593-625. | MR 2362700 | Zbl 1152.60316

[3] P. Brémaud and L. Massoulié. Stability of nonlinear Hawkes processes. Ann. Probab. 24 (1996) 1563-1588. | MR 1411506 | Zbl 0870.60043

[4] D. J. Daley and D. Vere-Jones. An Introduction to the Theory of Point Processes, 1st edition. Springer, New York, 1988. | MR 950166 | Zbl 1026.60061

[5] A. Dembo and O. Zeitouni. Large Deviations Techniques and Applications, 2nd edition. Springer, New York, 1998. | MR 1619036 | Zbl 1177.60035

[6] M. D. Donsker and S. R. S. Varadhan. Asymptotic evaluation of certain Markov process expectations for large time. IV. Comm. Pure Appl. Math. 36 (1983) 183-212. | MR 690656 | Zbl 0512.60068

[7] J. Grandell. Point processes and random measures. Adv. in Appl. Probab. 9 (1977) 502-526. | MR 478331 | Zbl 0376.60058

[8] A. G. Hawkes. Spectra of some self-exciting and mutually exciting point processes. Biometrika 58 (1971) 83-90. | MR 278410 | Zbl 0219.60029

[9] T. Liniger. Multivariate Hawkes processes. Ph.D. thesis, ETH, 2009. | Zbl 1242.62093

[10] R. S. Lipster and A. N. Shiryaev. Statistics of Random Processes II: Applications, 2nd edition. Springer, Berlin, 2001. | MR 1800858 | Zbl 1008.62073

[11] G. Stabile and G. L. Torrisi. Risk processes with non-stationary Hawkes arrivals. Methodol. Comput. Appl. Probab. 12 (2010) 415-429. | MR 2665268 | Zbl 1231.91239

[12] S. R. S. Varadhan. Special invited paper: Large deviations. Ann. Probab. 36 (2008) 397-419. | Zbl 1146.60003

[13] S. R. S. Varadhan. Large Deviations and Applications. SIAM, Philadelphia, 1984. | MR 758258 | Zbl 0549.60023

[14] L. Zhu. Large deviations for Markovian nonlinear Hawkes processes. Preprint, 2011. Available at arXiv:1108.2432. | MR 3313748

[15] L. Zhu. Central limit theorem for nonlinear Hawkes processes. J. Appl. Probab. 50 (2013) 760-771. | MR 3102513 | Zbl pre06216057