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

Résumé

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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Dates et versions

hal-02513614 , version 1 (20-03-2020)
hal-02513614 , version 2 (30-03-2022)

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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-02513614v1⟩
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