Non-linear Hawkes processes with memory kernels given by the sum of Erlang kernels are considered. It is shown that their stability properties can be studied in terms of an associated class of piecewise deterministic Markov processes, called Markovian cascades of successive memory terms. Explicit conditions implying the positive Harris recurrence of these processes are presented. The proof is based on integration by parts with respect to the jump times. A crucial property is the non-degeneracy of the transition semigroup which is obtained thanks to the invertibility of an associated Vandermonde matrix. For Lipschitz continuous rate functions we also show that these Markovian cascades converge to equilibrium exponentially fast with respect to the Wasserstein distance. Finally, an extension of the classical thinning algorithm is proposed to simulate such Markovian cascades.
Accepté le :
DOI : 10.1051/ps/2019005
Mots-clés : Hawkes processes, Erlang kernels, piecewise deterministic Markov process (PDMP), longtime behaviour, coupling
@article{PS_2019__23__770_0,
author = {Duarte, Aline and L\"ocherbach, Eva and Ost, Guilherme},
title = {Stability, convergence to equilibrium and simulation of non-linear {Hawkes} processes with memory kernels given by the sum of {Erlang} kernels},
journal = {ESAIM: Probability and Statistics},
pages = {770--796},
publisher = {EDP-Sciences},
volume = {23},
year = {2019},
doi = {10.1051/ps/2019005},
mrnumber = {4044609},
zbl = {1506.60051},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps/2019005/}
}
TY - JOUR AU - Duarte, Aline AU - Löcherbach, Eva AU - Ost, Guilherme TI - Stability, convergence to equilibrium and simulation of non-linear Hawkes processes with memory kernels given by the sum of Erlang kernels JO - ESAIM: Probability and Statistics PY - 2019 SP - 770 EP - 796 VL - 23 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2019005/ DO - 10.1051/ps/2019005 LA - en ID - PS_2019__23__770_0 ER -
%0 Journal Article %A Duarte, Aline %A Löcherbach, Eva %A Ost, Guilherme %T Stability, convergence to equilibrium and simulation of non-linear Hawkes processes with memory kernels given by the sum of Erlang kernels %J ESAIM: Probability and Statistics %D 2019 %P 770-796 %V 23 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps/2019005/ %R 10.1051/ps/2019005 %G en %F PS_2019__23__770_0
Duarte, Aline; Löcherbach, Eva; Ost, Guilherme. Stability, convergence to equilibrium and simulation of non-linear Hawkes processes with memory kernels given by the sum of Erlang kernels. ESAIM: Probability and Statistics, Tome 23 (2019), pp. 770-796. doi : 10.1051/ps/2019005. http://www.numdam.org/articles/10.1051/ps/2019005/
[1] , and , Mesures invariantes des processus de Markov récurrents. Sém. Proba. III, Vol. 88 of Lecture Notes in Mathematics. Springer, Berlin (1969) 24–33. | DOI | Numdam | MR | Zbl
[2] , , and , Qualitative properties of certain piecewise deterministic Markov processes. Ann. Inst. Henri Poincaré Probab. Statist. 51 (2015) 1040–1075. | DOI | Numdam | MR | Zbl
[3] , Separability and completeness for the wasserstein distance. Sém. Proba. XLI, Vol. 1934 of Lecture Notes in Mathematics. Springer, Berlin (2008) 371–377. | DOI | MR | Zbl
[4] and , Stability of nonlinear Hawkes processes. Ann. Probab. 24 (1996) 1563–1588. | DOI | MR | Zbl
[5] , Mean-field limit of generalized Hawkes processes. Stoch. Proc. Appl. 127 (2017) 3870–3912. | DOI | MR | Zbl
[6] , , and , Microscopic approach of a time elapsed neural model. Math. Models Methods Appl. Sci. 25 (2015) 2669–2719. | DOI | MR | Zbl
[7] and , Exact simulation of Hawkes process with exponentially decaying intensity. Electron. Commun. Probab. 18 (2013) 1–13. | DOI | MR | Zbl
[8] , and , Hawkes processes on large networks. Ann. App. Probab. 26 (2016) 216–261. | MR | Zbl
[9] and , Multi-class oscillating systems of interacting neurons. Stoch. Proc. Appl. 127 (2017) 1840–1869. | DOI | MR | Zbl
[10] , and , Parameter estimation in a stochastic model of the tubuloglomerular feedback mechanism in a rat nephron. Math. Biosci. 194 (2005) 49–69. | DOI | MR | Zbl
[11] , and , Exponential and uniform ergodicity of Markov processes. Ann. Probab. 23 (1995) 1671–1691. | DOI | MR | Zbl
[12] , and , Lasso and probabilistic inequalities for multivariate point processes. Bernoulli 21 (2015) 83–143. | DOI | MR | Zbl
[13] , Spectra of some self-exciting and mutually exciting point Processes. Biometrika, 58 (1971) 83–90. | DOI | MR | Zbl
[14] and , A cluster process representation of a self-exciting process. J. Appl. Probab. 11 (1974) 93–503. | DOI | MR | Zbl
[15] , Multivariate Point Processes: Predictable Projection, Radon-Nikodym Derivatives, Representation of Martingales. Z. Wahrscheinlichkeitstheorie verw. Gebiete 31 (1975) 235–253. | DOI | MR | Zbl
[16] , Approximation with sums of exponentials in lp[0, ∞). J. Approx. Theory 16 (1976) 384–408. | DOI | MR | Zbl
[17] , Convergence to equilibrium for time inhomogeneous jump diffusions with state dependent jump intensity. Preprint (2017). | arXiv | MR
[18] and , On nummelin splitting for continuous time harris recurrent Markov processes and application to kernel estimation for multi-dimensional diffusions. Stoch. Process. Appl. 118 (2008) 1301–1321. | DOI | MR | Zbl
[19] and , Stability of Markovian processes III : Foster-Lyapunov criteria for continuous-time processes. Adv. Appl. Probab. 25 (1993) 487–548. | MR | Zbl
[20] , Probability Metrics and the Stability of Stochastic Models. John Wiley and Sons, Chichester, USA (1991). | MR | Zbl
[21] , and , Age dependent Hawkes process. Preprint (2018). | arXiv
[22] and , The effect of different forms for the delay in a model of the nephron. Math. Biosci. Eng. 2 (2005) 97–109. | DOI | MR | Zbl
[23] , Large deviations for Markovian nonlinear Hawkes processes. Ann. Appl. Probab. 25 (2015) 548–581. | MR | Zbl
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