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://archive.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://archive.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://archive.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://archive.numdam.org/articles/10.1051/ps/2019005/
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