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

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.

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.

DOI : 10.1214/12-AIHP532
Classification : 60G55, 60F10
Mots-clés : large deviations, rare events, point processes, Hawkes processes, self-exciting processes 
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Zhu, Lingjiong. Process-level large deviations for nonlinear Hawkes point processes. Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 3, pp. 845-871. doi : 10.1214/12-AIHP532. http://www.numdam.org/articles/10.1214/12-AIHP532/

[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

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

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

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

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

[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 | Zbl

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

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

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

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

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

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

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

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

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

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