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Diffusion approximation of multi-class Hawkes processes: Theoretical and numerical analysis

Part of: Markov processes Probabilistic methods, simulation and stochastic differential equations Stochastic processes Stochastic analysis

Published online by Cambridge University Press:  08 October 2021

Julien Chevallier Open the ORCID record for Julien Chevallier [Opens in a new window] ,
Anna Melnykova Open the ORCID record for Anna Melnykova [Opens in a new window]  and
Irene Tubikanec Open the ORCID record for Irene Tubikanec [Opens in a new window]
Julien Chevallier*
Affiliation:
Université Grenoble Alpes
Anna Melnykova*
Affiliation:
Université de Cergy-Pontoise and Université Grenoble Alpes
Irene Tubikanec*
Affiliation:
Johannes Kepler University Linz
*
*Postal address: Université Grenoble Alpes, LJK UMR-CNRS 5224.
**Postal address: Université de Cergy-Pontoise, AGM UMR-CNRS 8088.
***Postal address: Institute for Stochastics, Johannes Kepler University Linz.
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Abstract

Oscillatory systems of interacting Hawkes processes with Erlang memory kernels were introduced by Ditlevsen and Löcherbach (Stoch. Process. Appl., 2017). They are piecewise deterministic Markov processes (PDMP) and can be approximated by a stochastic diffusion. In this paper, first, a strong error bound between the PDMP and the diffusion is proved. Second, moment bounds for the resulting diffusion are derived. Third, approximation schemes for the diffusion, based on the numerical splitting approach, are proposed. These schemes are proved to converge with mean-square order 1 and to preserve the properties of the diffusion, in particular the hypoellipticity, the ergodicity, and the moment bounds. Finally, the PDMP and the diffusion are compared through numerical experiments, where the PDMP is simulated with an adapted thinning procedure.

Keywords

Piecewise deterministic Markov processesHawkes processesstochastic differential equationsdiffusion processesneuronal modelsnumerical splitting schemes

MSC classification

Primary: 60H35: Computational methods for stochastic equations 65C20: Models, numerical methods
Secondary: 65C30: Stochastic differential and integral equations 60G55: Point processes 60J25: Continuous-time Markov processes on general state spaces
Type
Original Article
Information
Advances in Applied Probability , Volume 53 , Issue 3 , September 2021 , pp. 716 - 756
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Copyright
© The Author(s) 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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