Large deviations for the mean-field limit of Hawkes processes

. Hawkes processes are a class of simple point processes whose intensity depends on the past history, and is in general non-Markovian. Limit theorems for Hawkes processes in various asymptotic regimes have been studied in the literature. In this paper, we study a multidimensional nonlinear Hawkes process in the asymptotic regime when the dimension goes to infinity, whose mean-field limit is a time-inhomogeneous Poisson process, and our main result is a large deviation principle for the mean-field limit.

[1]  Zo'e Agathe-Nerine.  Multivariate Hawkes processes on inhomogeneous random graphs , 2021, 2106.12259.

[2]  C. Pouzat,et al.  Interacting Hawkes processes with multiplicative inhibition , 2021, Stochastic Processes and their Applications.

[3]  S. Rotter,et al.  Mean-field limits for non-linear Hawkes processes with excitation and inhibition , 2021, Stochastic Processes and their Applications.

[4]  S. Ditlevsen,et al.  Stability and mean-field limits of age dependent Hawkes processes , 2020, Annales de l'Institut Henri Poincaré, Probabilités et Statistiques.

[5]  Valentin Schmutz.  Mean-field limit of age and leaky memory dependent Hawkes processes , 2020, Stochastic Processes and their Applications.

[6]  W. Stannat,et al.  Fluctuation limits for mean-field interacting nonlinear Hawkes processes , 2019, 1912.10854.

[7]  C. Graham.  Regenerative properties of the linear hawkes process with unbounded memory , 2019, 1905.11053.

[8]  Eva Locherbach,et al.  Mean field limits for interacting Hawkes processes in a diffusive regime , 2019, Bernoulli.

[9]  Nicolas Privault,et al.  Functional inequalities for marked point processes , 2019, Electronic Journal of Probability.

[10]  Manon Costa,et al.  Renewal in Hawkes processes with self-excitation and inhibition , 2018, Advances in Applied Probability.

[11]  A. Pascucci,et al.  Systemic risk in a mean-field model of interbank lending with self-exciting shocks , 2017, 1710.00231.

[12]  E. Locherbach.  Large deviations for cascades of diffusions arising in oscillating systems of interacting Hawkes processes , 2017, 1709.09356.

[13]  Julien Chevallier,et al.  Stimulus Sensitivity of a Spiking Neural Network Model , 2017, 1709.06958.

[14]  G. Torrisi.  Poisson approximation of point processes with stochastic intensity, and application to nonlinear Hawkes processes , 2017 .

[15]  G. Ost,et al.  Mean field limits for nonlinear spatially extended Hawkes processes with exponential memory kernels , 2017, Stochastic Processes and their Applications.

[16]  Lingjiong Zhu,et al.  Some asymptotic results for nonlinear Hawkes processes , 2017, Stochastic Processes and their Applications.

[17]  Lingjiong Zhu,et al.  Precise deviations for Hawkes processes , 2017, Bernoulli.

[18]  Julien Chevallier.  Fluctuations for mean-field interacting age-dependent Hawkes processes , 2016, 1611.02008.

[19]  G. Torrisi.  Gaussian approximation of nonlinear Hawkes processes , 2016, 1609.08899.

[20]  Xuefeng Gao,et al.  Functional central limit theorems for stationary Hawkes processes and application to infinite-server queues , 2016, Queueing Systems.

[21]  Lingjiong Zhu,et al.  Large deviations and applications for Markovian Hawkes processes with a large initial intensity , 2016, Bernoulli.

[22]  S. Delattre,et al.  HAWKES PROCESSES ON LARGE NETWORKS , 2016 .

[23]  S. Ditlevsen,et al.  Multi-class oscillating systems of interacting neurons , 2015, 1512.00265.

[24]  Julien Chevallier.  Mean-field limit of generalized Hawkes processes , 2015, 1510.05620.

[25]  S. Delattre,et al.  Statistical inference versus mean field limit for Hawkes processes , 2015, 1507.02887.

[26]  Mathieu Rosenbaum,et al.  Rough fractional diffusions as scaling limits of nearly unstable heavy tailed Hawkes processes , 2015, 1504.03100.

[27]  Mathieu Rosenbaum,et al.  Limit theorems for nearly unstable Hawkes processes , 2013, 1310.2033.

[28]  Lingjiong Zhu.  Nonlinear Hawkes Processes , 2013, 1304.7531.

[29]  Niels Richard Hansen,et al.  Exponential Martingales and Changes of Measure for Counting Processes , 2012, 1205.2244.

[30]  Lingjiong Zhu,et al.  Central Limit Theorem for Nonlinear Hawkes Processes , 2012, Journal of Applied Probability.

[31]  Emmanuel Bacry,et al.  Scaling limits for Hawkes processes and application to financial statistics , 2012, 1202.0842.

[32]  Lingjiong Zhu,et al.  Large deviations for Markovian nonlinear Hawkes processes , 2011, 1108.2432.

[33]  Lingjiong Zhu.  Process-level large deviations for nonlinear Hawkes point processes , 2011, 1108.2431.

[34]  I. Grigorescu.  Large deviations for a catalytic Fleming‐Viot branching system , 2007 .

[35]  C. Bordenave,et al.  Large Deviations of Poisson Cluster Processes , 2007, 0704.1463.

[36]  T. Kurtz,et al.  Large Deviations for Stochastic Processes , 2006 .

[37]  C. Villani.  Topics in Optimal Transportation , 2003 .

[38]  A. Guillin,et al.  Large and Moderate Deviations for Estimators of Quadratic Variational Processes of Diffusions , 1999 .

[39]  Amir Dembo,et al.  Large Deviations Techniques and Applications , 1998 .

[40]  P. Brémaud,et al.  STABILITY OF NONLINEAR HAWKES PROCESSES , 1996 .

[41]  Stefano Olla,et al.  Hydrodynamics and large deviation for simple exclusion processes , 1989 .

[42]  George Papanicolaou,et al.  Nonlinear diffusion limit for a system with nearest neighbor interactions , 1988 .

[43]  A. Hawkes,et al.  A cluster process representation of a self-exciting process , 1974, Journal of Applied Probability.

[44]  A. Hawkes.  Spectra of some self-exciting and mutually exciting point processes , 1971 .

[45]  Francis Comets,et al.  Large Deviations and Applications , 2011, International Encyclopedia of Statistical Science.

[46]  M. Yor,et al.  Continuous martingales and Brownian motion , 1990 .

[47]  A. Shiryaev,et al.  Limit Theorems for Stochastic Processes , 1987 .