A Hawkes Microfoundation for Multitype Inverse Gaussian Subordinators

Paper: arXiv 2610.10525

Authors: Yingli Wang, Wei Xu, Lingjiong Zhu

Abstract

We provide an event-level Hawkes microfoundation for a multitype inverse-Gaussian stochastic clock. We show that the event counts and integrated intensities of nearly critical multivariate linear Hawkes processes converge jointly to a multivariate pure-jump subordinator when reproduction delays have finite mean and immigration is balanced so that rare large families remain visible. The dependence among the limiting coordinates is inherited from microscopic cross-excitation, and the limit admits a Brownian additive-field first-passage representation. We call this process the multitype inverse-Gaussian subordinator. Its diagonal case consists of independent classical inverse-Gaussian subordinators, with the univariate model as a further special case. The square-root specialization recovers the inverse-Gaussian clock obtained at the boundary of hyper-rough square-root models by Abi Jaber–Attal–Rosenbaum (\textit{Ann. Appl. Probab.} \textbf{36}(4): 3635–3660, 2026). Our cluster proof exposes the underlying mechanism: finite-variance near-critical branching creates rare but macroscopic families, while their internal timing disappears on the observation scale, so each family becomes a jump. We also clarify the relation with the multivariate Hawkes scaling theory of Xu (arXiv:2412.14459): its diagonal-atom condition yields the diagonal specialization, whereas the genuinely coupled limit requires a separate uniqueness argument. In addition, under a common tilted-stability condition, we replace the high-intensity assumption in that theory by a weaker accumulated-activity condition and provide the Riccati identification needed when the potential has an atom. We strengthen convergence of the count and compensator and extend the scalar conclusion to finite-variance age-dependent branching clusters.

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents highly complex mathematical derivations for a novel multitype inverse-Gaussian subordinator. While theoretically rich, it lacks empirical validation or practical backtesting, focusing purely on theoretical convergence and microfoundations. The clarity is moderate, given the highly specialized nature of the topic.

Research Flowchart

  flowchart TD
    A[Research Goal: Event-Level Hawkes Microfoundation for Multitype Inverse-Gaussian Subordinator] --> B{Methodology: Scaling Limits of Hawkes Processes};
    B --> C{Inputs: Nearly Critical MV Linear Hawkes Process, Reproduction Delays, Balanced Immigration};
    C --> D{Computational Process: Joint Convergence Analysis, Cluster Proof, Uniqueness Arguments};
    D --> E[Outcomes: Multitype Inverse-Gaussian Subordinator (IGS) Defined];
    E --> F[Key Findings:
                - IGS properties (Brownian additive-field representation)
                - Diagonal & Square-root specializations
                - Mechanism: finite-variance near-critical branching -> jumps
                - Relation to Xu's scaling theory clarified
                - Strengthened convergence & extended scalar conclusion];

Cited References (49)

1. From Hyper Roughness to Jumps as $H\to-1/2$ (2026)
2. Volterra Clocks and Their Pure-Jump Limits: Hitting Times of Curved Boundaries (2026) — link
3. Efficient Simulation of Hawkes Processes Using Their Affine Volterra Structure (2025) — link
4. Reconciling Rough Volatility with Jumps — → 2303.07222
5. Affine Volterra Processes (2019)
6. Weak Existence and Uniqueness for Affine Stochastic Volterra Equations with $L^1$ -Kernels (2021)
7. Limits and Approximations for the Busy-Period Distribution in Single-Server Queues (1995)
8. Hawkes Processes in Finance (2015)

5 of 49 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.