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Theoretical analysis and simulation methods for Hawkes processes and their diffusion approximation

Abstract : Oscillatory systems of interacting Hawkes processes with Erlang memory kernels were introduced in Ditlevsen and Löcherbach (2017). They are piecewise deterministic Markov processes (PDMP) and can be approximated by a stochastic diffusion. 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 ergod-icity 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.
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https://hal.archives-ouvertes.fr/hal-02513614
Contributor : Anna Melnykova <>
Submitted on : Friday, March 20, 2020 - 6:23:59 PM
Last modification on : Saturday, March 28, 2020 - 1:13:52 AM

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  • HAL Id : hal-02513614, version 1

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Julien Chevallier, Anna Melnykova, Irene Tubikanec. Theoretical analysis and simulation methods for Hawkes processes and their diffusion approximation. 2020. ⟨hal-02513614⟩

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