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
@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},
     pages = {845--871},
     publisher = {Gauthier-Villars},
     volume = {50},
     number = {3},
     year = {2014},
     doi = {10.1214/12-AIHP532},
     mrnumber = {3224291},
     zbl = {1296.60129},
     language = {en},
     url = {http://www.numdam.org/articles/10.1214/12-AIHP532/}
}
TY  - JOUR
AU  - Zhu, Lingjiong
TI  - Process-level large deviations for nonlinear Hawkes point processes
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2014
SP  - 845
EP  - 871
VL  - 50
IS  - 3
PB  - Gauthier-Villars
UR  - http://www.numdam.org/articles/10.1214/12-AIHP532/
DO  - 10.1214/12-AIHP532
LA  - en
ID  - AIHPB_2014__50_3_845_0
ER  - 
%0 Journal Article
%A Zhu, Lingjiong
%T Process-level large deviations for nonlinear Hawkes point processes
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2014
%P 845-871
%V 50
%N 3
%I Gauthier-Villars
%U http://www.numdam.org/articles/10.1214/12-AIHP532/
%R 10.1214/12-AIHP532
%G en
%F 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, 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

Cité par Sources :