Fluctuation limits for mean-field interacting nonlinear Hawkes processes

Sophie Heesen and Wilhelm Stannat

Stochastic Processes and their Applications, 2021, vol. 139, issue C, 280-297

Abstract: We investigate the asymptotic behavior of networks of interacting non-linear Hawkes processes modeling a homogeneous population of neurons in the large population limit. In particular, we prove a functional central limit theorem for the mean spike-activity thereby characterizing the asymptotic fluctuations in terms of a stochastic Volterra integral equation. Our approach differs from previous approaches in making use of the associated resolvent in order to represent the fluctuations as Skorokhod continuous mappings of weakly converging martingales. Since the Lipschitz properties of the resolvent are explicit, our analysis in principle also allows to derive approximation errors in terms of driving martingales. We also discuss extensions of our results to multi-class systems.

Keywords: Hawkes process; Mean field limit; Skorokhod topology; Stochastic Volterra integral equation (search for similar items in EconPapers)
Date: 2021
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Citations: View citations in EconPapers (3)

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DOI: 10.1016/j.spa.2021.05.007

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