Diffusion approximation of multi-class Hawkes processes: Theoretical and numerical analysis. (September 2021)

• Record Type: Journal Article
• Authors: Chevallier, Julien; Melnykova, Anna; Tubikanec, Irene
• Is Part Of: Advances in applied probability. Volume 53:Number 3(2021)
• Page Start: 716
• Page End: 756
• Publication Date: 2021-09
• Subjects: Piecewise deterministic Markov processes -- Hawkes processes -- stochastic differential equations -- diffusion processes -- neuronal models -- numerical splitting schemes; 60H35 -- 65C20 -- 65C30 -- 60G55 -- 60J25; Probabilities -- Periodicals; Stochastic models -- Periodicals; Electronic journals; Periodicals; 519.2
• DOI: 10.1017/apr.2020.73
• Languages: English
• ISSNs: 0001-8678

Abstract

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.