Paper: arXiv 2501.11648

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.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper is highly mathematical, featuring advanced stochastic calculus, mean-field theory, and stochastic Volterra equations with intricate scaling limits and propagation of chaos results. It is purely theoretical with no empirical data, backtests, or implementation details, focusing entirely on establishing mathematical scaling limits for nearly unstable Hawkes processes.

Research Flowchart

  flowchart TD
  A["Research Goal<br>Establish scaling limits for nearly<br>unstable Hawkes processes"] --> B["Methodology: Extension of<br>Jaisson & Rosenbaum approach"]
  B --> C{"Asymptotic Criticality Condition<br>‖φ<sup>n</sup>‖<sub>L<sup>1</sup></sub> → 1"}
  C --> D["Computational Scaling Limits<br>Stochastic Volterra Diffusions"]
  D --> E["Propagations of Chaos<br>Mean-field interaction analysis"]
  E --> F["Key Findings: Three Regimes<br>Based on n(1-‖φ<sup>n</sup>‖<sub>L<sup>1</sup></sub>)<sup>2</sup> asymptotics"]