Volume 23, 2019
|Page(s)||770 - 796|
|Published online||20 December 2019|
Stability, convergence to equilibrium and simulation of non-linear Hawkes processes with memory kernels given by the sum of Erlang kernels
Universidade de São Paulo,
São Paulo, Brazil.
2 Université de Paris 1 Panthéon Sorbonne, Paris, France.
3 Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil.
* Corresponding author: firstname.lastname@example.org
Accepted: 28 March 2019
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.
Mathematics Subject Classification: 60G55 / 60J75 / 60K10
Key words: Hawkes processes / Erlang kernels / piecewise deterministic Markov process (PDMP) / longtime behaviour / coupling
© EDP Sciences, SMAI 2019
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