Mean-Field Limits for Nearly Unstable Hawkes Processes
Mean-Field Limits for Nearly Unstable Hawkes Processes ArXiv ID: 2501.11648 “View on arXiv” Authors: Unknown 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 ${“φ^n"}$, specifically $|φ^n|{“L^1”} \to 1$, 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 $n(1-|φ^n|{“L^1”})^2$. These results provide a significant generalization of the findings by Delattre, Fournier and Hoffmann. ...