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A Hawkes Microfoundation for Multitype Inverse Gaussian Subordinators

This paper establishes that nearly critical multivariate linear Hawkes processes with finite-mean reproduction delays converge to a multitype inverse-Gaussian subordinator, providing a rigorous event-level microfoundation where rare, macroscopic branching families manifest as jumps in a stochastic clock that generalizes classical inverse-Gaussian models and connects to hyper-rough volatility frameworks.

Original authors: Yingli Wang, Wei Xu, Lingjiong Zhu

Published 2026-10-08

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Yingli Wang, Wei Xu, Lingjiong Zhu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the bustling world of high-frequency finance, markets do not move in a smooth, continuous stream. Instead, they pulse with a chaotic rhythm of individual trades, each one triggering a cascade of reactions from other participants. This phenomenon, known as self-excitation, means that a single event makes future events more likely, creating clusters of activity that can look like sudden avalanches. For decades, mathematicians have tried to model this behavior using tools that treat these clusters as distinct families of events, where an initial "immigrant" arrival sparks a lineage of "offspring" trades. Understanding how these microscopic clusters behave when they grow large is crucial for predicting macroscopic market volatility, yet the transition from the chaotic noise of individual trades to a stable, predictable pattern has remained a difficult puzzle.

A team of researchers has now solved a specific and critical piece of this puzzle by showing exactly how these chaotic clusters collapse into a new, predictable form of randomness. They focused on a scenario where the market is "nearly critical," meaning it is balanced on a knife-edge where a single trade could theoretically trigger an infinite chain reaction, but in reality, the chains are just very long and rare. By observing these systems over a long period and scaling up the time, the researchers demonstrated that the chaotic, step-by-step accumulation of thousands of tiny trades does not smooth out into a gentle curve. Instead, it collapses into a series of sudden, massive jumps. They proved that the timing and size of these jumps follow a specific statistical law known as the inverse Gaussian distribution, a pattern that had previously only been glimpsed through complex, indirect mathematical limits.

The researchers achieved this by building a detailed model of how these event families interact. They imagined a system where different types of events can excite one another, creating a web of cross-dependencies. When they zoomed out to look at the system as a whole, they found that the internal timing of the events within a single family—the precise moments when a trade triggers another trade a split second later—disappears entirely. On the large scale, the entire family of events, no matter how complex its internal structure, behaves as a single, instantaneous unit. It is as if the entire history of a massive, multi-generational family of trades is compressed into a single heartbeat. This compression happens because the time it takes for these families to play out is finite on average, and when the system is scaled up, this duration becomes negligible compared to the observation window.

What remains is a process where the total volume of activity arrives in distinct, unpredictable bursts. The researchers showed that the size and frequency of these bursts are governed by a simple rule: the more intense the initial "immigration" of new events, and the closer the system is to the critical tipping point, the larger and more frequent these jumps become. Crucially, they proved that the dependence between different types of events—how a trade in one sector might trigger a reaction in another—is preserved in this new, simplified form. The complex web of interactions at the microscopic level transforms into a structured relationship between the sizes of the jumps at the macroscopic level. This means that the new model does not just describe random noise; it captures the specific way different market forces are linked together.

The study also clarified why previous attempts to model this behavior sometimes failed or required overly strict assumptions. Earlier theories suggested that for these patterns to emerge, the intensity of the events had to be extremely high. The researchers demonstrated that this was not necessary. Instead, the key factor is the total accumulated activity over time. As long as enough events occur to form a substantial history, the specific pattern of jumps emerges naturally, regardless of how fast the individual events are arriving. This finding broadens the applicability of the model, showing that it describes a fundamental property of near-critical systems rather than a rare edge case.

Furthermore, the team provided a clear physical interpretation for the mathematical results. They connected the abstract statistical law to a concrete mechanism involving a "random walk" that drifts downward until it hits a barrier. In their model, the total size of a family of events corresponds to the time it takes for this random walk to reach zero. This connection allows them to visualize the complex branching process as a simple journey of a particle moving through time, where the moment it hits the ground determines the size of the resulting jump. This bridge between the microscopic branching structure and the macroscopic jump process confirms that the inverse Gaussian clock is not an arbitrary mathematical invention but a natural consequence of how finite-variance branching systems behave when pushed to their limits.

The implications of this work extend beyond pure theory. By identifying the exact conditions under which these jumps occur, the researchers have provided a robust framework for understanding market volatility. They showed that even when the underlying rules of the market are complex and involve many interacting types of events, the resulting large-scale behavior is surprisingly simple and predictable. The model reveals that the "memory" of the system—the way past events influence the future—is not lost but is instead encoded in the size and timing of the jumps. This offers a new way to think about financial risk, suggesting that the most significant market movements are not the result of a slow, grinding accumulation of pressure, but rather the sudden release of energy from rare, massive families of events that have been building up in the background.

In the end, this research provides a clear map of the transition from chaos to order. It shows that when a system is balanced on the edge of instability, the noise of individual events does not wash out into a blur. Instead, it organizes itself into a distinct, jump-driven rhythm. The researchers have proven that this rhythm follows a specific, mathematically precise law, one that preserves the essential connections between different parts of the system. Their work confirms that the inverse Gaussian distribution is the natural language for describing these critical moments, offering a powerful tool for anyone trying to understand the hidden structure of complex, self-exciting systems.

Technical Summary: A Hawkes Microfoundation for Multitype Inverse Gaussian Subordinators

Problem Statement
The paper addresses the derivation of a rigorous microfoundation for multitype inverse-Gaussian (IG) subordinators from nearly critical multivariate linear Hawkes processes. While previous literature has established connections between Hawkes processes and continuous stochastic volatility models (e.g., CIR) or rough volatility limits (fractional kernels), the specific regime yielding a pure-jump IG clock—particularly in a multivariate setting with coupled cross-excitation—requires clarification. The authors aim to identify the precise scaling conditions under which event counts and integrated intensities of a multivariate Hawkes system converge jointly to a multitype IG subordinator, characterized by a Brownian additive-field first-passage representation.

Methodology
The authors employ a cluster-based approach rooted in the immigration–branching representation of Hawkes processes. The methodology proceeds through several key technical steps:

Scaling Regime: The analysis focuses on a "finite-mean calendar-collapse" regime. The branching ratio $a_T$ $a_{T}$ approaches 1 such that $r_T = 1 - a_T \to 0$ $r_{T} = 1 - a_{T} \to 0$ . The scaling assumes:
- $T r_T \to \infty$ (ensuring the internal timing of families collapses).
- $T \mu_T r_T \to \eta$ (balancing rare, large families to remain visible).
- The kernel mass matrix $A_T$ satisfies $(I - A_T)/r_T \to B$ , where $B$ is a nonsingular M-matrix.
Cluster Decomposition: The Hawkes process is decomposed into immigrant families. The authors analyze the total progeny of these families, showing that under the specified scaling, families with finite-variance offspring produce macroscopic masses of order $r_T^{-2}$ .
Collapse Mechanism: A critical technical component is proving that the internal calendar time of these large families vanishes in the limit. Using matrix renewal estimates and expected age measures, the authors demonstrate that the difference between the actual Hawkes count and a "collapsed" compound Poisson process (where all descendants are placed at the immigrant time) converges to zero in $L^1$ and probability, provided $T r_T \to \infty$ .
Topological Convergence: The convergence is established in the product $M_1$ topology on the space of càdlàg functions. The authors argue that the $J_1$ topology is insufficient because the limit process has jumps, while the pre-limit processes consist of many small, rapid jumps (microscopic staircases) that $M_1$ can merge into single macroscopic jumps.
Riccati Identification: The paper refines general multivariate Hawkes scaling theory (extending Xu, arXiv:2412.14459) by replacing the high-intensity assumption ( $\theta_n \to \infty$ ) with a weaker accumulated-activity condition ( $n \theta_n \to \infty$ ). This involves establishing local $L^p$ convergence of Riccati-Volterra solutions and handling cases where the potential measure has an atom at zero.

Key Contributions and Results

Definition of the Multitype IG Subordinator:
The paper defines a multitype IG subordinator $Z$ via its Laplace exponent, determined by the unique non-negative solution $z(u)$ to the coupled quadratic system:
$B^\top z + \frac{1}{2} z^{\circ 2} = u$
where $z^{\circ 2}$ denotes the component-wise square. This process is shown to admit a representation as the first-passage time of a Brownian additive field $X_s = (W_1(s_1), \dots, W_d(s_d))^\top - Bs$ to level $-x$ .
Joint Convergence Theorem (Theorem 2.2):
Under the stated assumptions, the rescaled Hawkes count process $X^T_t = r_T^2 N^T_{Tt}$ and the rescaled compensator $C^T_t = r_T^2 \Lambda^T_{Tt}$ converge jointly to $(Z, Z)$ in the product $M_1$ topology.
- Dependence Structure: The cross-dependence in the limit is inherited directly from the microscopic cross-excitation matrix $B$ . If $B$ is diagonal, the limit consists of independent classical IG subordinators.
- Topological Necessity: The paper proves that marginal convergence in the $J_1$ topology is impossible for coordinates where the limit has jumps, as the pre-limit paths are continuous (compensators) or have vanishing jump sizes ( $r_T^2$ ).
Refinement of General Scaling Theory (Section 3):
The authors extend the general multivariate Hawkes scaling theory of Xu (arXiv:2412.14459). They show that under a common tilted-stability condition and a diagonal-atom assumption, the requirement $\theta_n \to \infty$ can be relaxed to $n \theta_n \to \infty$ . This allows the theory to cover the "calendar-collapse" region where $T r_T^2 \to \infty$ or oscillates, which was previously excluded. They also provide a uniqueness argument for the Riccati equation when the potential has an atom.
Exact Calendar-Collapse Condition (Lemma 5.1):
For a fixed scalar reproduction-time density, the paper proves that the calendar collapse (where the internal timing of a family disappears) occurs at a fixed time $t$ if and only if $T r_T \to \infty$ . If $T r_T$ remains bounded, the family depth remains macroscopic, leading to different limits (e.g., integrated CIR).
Universality and Extensions (Section 6):
The results are extended to age-dependent branching clusters with finite-variance offspring distributions (not necessarily Poisson). The limiting IG class persists, with the offspring variance $\sigma^2$ modifying the dispersion parameter of the IG clock but leaving the mean activity rate determined by the immigration balance.

Significance and Claims
The paper claims to provide a direct event-level microfoundation for the multitype inverse-Gaussian subordinator, bridging the gap between microscopic self-exciting point processes and macroscopic pure-jump stochastic clocks.

Mechanism Clarification: The work isolates the mechanism: finite-variance near-critical branching creates rare but macroscopic families; the balance of immigration keeps these families visible; and the finite-mean delay assumption causes their internal timing to collapse, turning each family into a single jump.
Dependence Inheritance: Unlike arbitrary dependence constructions, the multivariate limit's dependence structure is explicitly shown to arise from the microscopic cross-excitation kernel.
Unification of Limits: The diagonal specialization recovers the scalar IG limit, and the square-root normalization recovers the hyper-rough boundary limit found in Volterra-based approaches (Abi Jaber–Attal–Rosenbaum, 2026), demonstrating that these distinct mathematical routes converge to the same probabilistic object.
Topological Precision: The paper emphasizes the necessity of the $M_1$ topology for capturing the convergence of monotone processes with vanishing jump sizes to jump limits, correcting potential misinterpretations that might arise from using the $J_1$ topology.

The authors position this work as a rigorous completion of the "rough volatility" program's boundary cases, specifically identifying the conditions under which Hawkes processes generate inverse-Gaussian clocks rather than rough fractional or continuous diffusion limits.

Technical Summary: A Hawkes Microfoundation for Multitype Inverse Gaussian Subordinators

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