Mean-Field Limits for Nearly Unstable Hawkes Processes

TL;DR

This paper derives scaling limits for nearly unstable Hawkes processes in a mean-field setting, showing they converge to stochastic Volterra diffusions and identifying different regimes based on kernel asymptotics.

Contribution

It extends existing methods to establish general scaling limits and propagation of chaos for nearly unstable Hawkes processes with mean-field interactions.

Findings

Hawkes processes converge to stochastic Volterra diffusions under criticality.

Three regimes identified depending on kernel asymptotics.

Results generalize previous work by Delattre, Fournier, and Hoffmann.

Abstract

In this paper, we establish general scaling limits for nearly unstable Hawkes processes in a mean-field regime by extending the method introduced by Jaisson and Rosenbaum. Under a mild asymptotic criticality condition on the self-exciting kernels $\{\phi^n\}$, specifically $\|\phi^n\|_{L^1} \to 1$, we first show that the scaling limits of these Hawkes processes are necessarily stochastic Volterra diffusions of affine type. Moreover, we establish a propagation of chaos result for Hawkes systems with mean-field interactions, highlighting three distinct regimes for the limiting processes, which depend on the asymptotics of $n(1-\|\phi^n\|_{L^1})^2$. These results provide a significant generalization of the findings by Delattre, Fournier and Hoffmann.