zbMATH — the first resource for mathematics

Process-level large deviations for nonlinear Hawkes point processes. (English. French summary) Zbl 1296.60129
Summary: 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.

60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60F10 Large deviations
Full Text: DOI Euclid arXiv
[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 . 1202.0842 · Zbl 1292.60032 · arxiv.org
[2] C. Bordenave and G. L. Torrisi. Large deviations of Poisson cluster processes. Stoch. Models 23 (2007) 593-625. · Zbl 1152.60316 · doi:10.1080/15326340701645959
[3] P. Brémaud and L. Massoulié. Stability of nonlinear Hawkes processes. Ann. Probab. 24 (1996) 1563-1588. · Zbl 0870.60043 · doi:10.1214/aop/1065725193
[4] D. J. Daley and D. Vere-Jones. An Introduction to the Theory of Point Processes , 1st edition. Springer, New York, 1988. · Zbl 0657.60069
[5] A. Dembo and O. Zeitouni. Large Deviations Techniques and Applications , 2nd edition. Springer, New York, 1998. · Zbl 0896.60013
[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. · Zbl 0512.60068 · doi:10.1002/cpa.3160360204
[7] J. Grandell. Point processes and random measures. Adv. in Appl. Probab. 9 (1977) 502-526. · Zbl 0376.60058 · doi:10.2307/1426111
[8] A. G. Hawkes. Spectra of some self-exciting and mutually exciting point processes. Biometrika 58 (1971) 83-90. · Zbl 0219.60029 · doi:10.1093/biomet/58.1.83
[9] T. Liniger. Multivariate Hawkes processes. Ph.D. thesis, ETH, 2009.
[10] R. S. Lipster and A. N. Shiryaev. Statistics of Random Processes II: Applications , 2nd edition. Springer, Berlin, 2001.
[11] G. Stabile and G. L. Torrisi. Risk processes with non-stationary Hawkes arrivals. Methodol. Comput. Appl. Probab. 12 (2010) 415-429. · Zbl 1231.91239 · doi:10.1007/s11009-008-9110-6
[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. · Zbl 0549.60023
[14] L. Zhu. Large deviations for Markovian nonlinear Hawkes processes. Preprint, 2011. Available at . 1108.2432 · arxiv.org
[15] L. Zhu. Central limit theorem for nonlinear Hawkes processes. J. Appl. Probab. 50 (2013) 760-771. · Zbl 1306.60015 · doi:10.1239/jap/1378401234
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.