Mean-Field Limits for Nearly Unstable Hawkes Processes

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 $\{\ddot{I}{•}^n\}$, specifically $\Vert \ddot{I}{•}^n{\Vert }_{L^1}\to 1$, 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 $n(1-\Vert \ddot{I}{•}^n{\Vert }_{L^1}{)}^2$. These results provide a significant generalization of the findings by Delattre, Fournier and Hoffmann.