The Malliavin-Stein method for Hawkes functionals

In this paper, following Nourdin-Peccati’s methodology, we combine the Malliavin calculus and Stein’s method to provide general bounds on the Wasserstein distance between functionals of a compound Hawkes process and a given Gaussian density. To achieve this, we rely on the Poisson embedding representation of an Hawkes process to provide a Malliavin calculus for the Hawkes processes, and more generally for compound Hawkes processes. As an application, we close a gap in the literature by providing the first Berry-Esséen bounds associated to Central Limit Theorems for the compound Hawkes process.

[1]  Caroline Hillairet,et al.  Multivariate Hawkes process for cyber insurance , 2020, Annals of Actuarial Science.

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

[3]  Guilherme Ost,et al.  Stability, convergence to equilibrium and simulation of non-linear Hawkes processes with memory kernels given by the sum of Erlang kernels , 2016, ESAIM: Probability and Statistics.

[4]  Caroline Hillairet,et al.  An expansion formula for Hawkes processes and application to cyber-insurance derivatives , 2021 .

[5]  Nicolas Privault,et al.  Stochastic Analysis in Discrete and Continuous Settings: With Normal Martingales , 2009 .

[6]  Characterization of infinite divisibility by duality formulas. Application to Lévy processes and random measures , 2013 .

[7]  J. Picard.  Formules de dualité sur l'espace de Poisson , 1996 .

[8]  Lingjiong Zhu.  Limit Theorems for a Cox-Ingersoll-Ross Process with Hawkes Jumps , 2014, J. Appl. Probab..

[9]  E. Bacry,et al.  Hawkes Processes in Finance , 2015, 1502.04592.

[10]  Murad S. Taqqu,et al.  Stein’s method and Normal approximation of Poisson functionals , 2008, 0807.5035.

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

[12]  Nicolas Privault.  Stein approximation for multidimensional Poisson random measures by third cumulant expansions , 2018, 1806.00235.

[13]  E. Bacry,et al.  Some limit theorems for Hawkes processes and application to financial statistics , 2013 .

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

[15]  G. Peccati,et al.  Stein’s method on Wiener chaos , 2007, 0712.2940.

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

[17]  Yosihiko Ogata,et al.  On Lewis' simulation method for point processes , 1981, IEEE Trans. Inf. Theory.

[18]  Ulrike Goldschmidt,et al.  An Introduction To The Theory Of Point Processes , 2016 .

[19]  Wei Xu,et al.  Functional limit theorems for marked Hawkes point measures , 2019, Stochastic Processes and their Applications.

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

[21]  C. Stein.  A bound for the error in the normal approximation to the distribution of a sum of dependent random variables , 1972 .

[22]  D. Nualart,et al.  Anticipative calculus for the Poisson process based on the Fock space , 1990 .

[23]  F. Utzet,et al.  Canonical Lévy process and Malliavin calculus , 2007 .

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

[25]  Kay Giesecke,et al.  Affine Point Processes and Portfolio Credit Risk , 2010, SIAM J. Financial Math..

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

[27]  Wei Xu,et al.  A Scaling Limit for Limit Order Books Driven by Hawkes Processes , 2017, SIAM J. Financial Math..