Mean-Field Limits for Nearly Unstable Hawkes Processes

TL;DR

This work establishes mean-field and scaling limits for nearly unstable Hawkes processes. Under a mild criticality condition $\|\phi^n\|_{L^1}\to1$, the scaling limits of multidimensional Hawkes processes converge to affine stochastic Volterra diffusions, with the derivative of the limit obeying a stochastic Volterra equation driven by Brownian noise. In the mean-field regime, with $\mu_i^n=\mu_0^n/n$ and $\phi^n_{ij}=\varphi^n/n$, the aggregate system exhibits a propagation of chaos with three regimes determined by $\zeta=\lim n\beta_n^2$: synchronization ($\zeta=0$), conditional independence ($\zeta\in(0,\infty)$), or extinction ($\zeta=\infty$). The results generalize prior work by relaxing kernel structure and connect the macroscopic limits to affine Volterra dynamics, providing a unifying framework for mean-field Hawkes in near-critical regimes and linking to applications in neuroscience and finance. The findings offer rigorous micro-to-macro descriptions and pave the way for analyzing fluctuations around these limits.

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.

Theorems & Definitions (27)

• Theorem 2.2
• Remark 2.3
• Lemma 2.4
• Proposition 2.5
• Theorem 3.1
• Theorem 3.2
• Remark 3.3
• Remark 3.4
• Lemma 4.1
• proof