Hawkes Excitation Norms Overview

• Hawkes excitation norms are defined as the L1 integral (or spectral radius in multivariate cases) of the kernel, quantifying the average number of offspring events.
• They serve as key indicators for process stationarity and stability, ensuring non-explosion by maintaining the norm below unity.
• These norms bridge micro-level event interactions with macroscopic phenomena, influencing applications from high-frequency finance to neuroscience and risk modeling.

A Hawkes excitation norm quantifies the cumulative impact of prior events on the instantaneous intensity of a Hawkes process via the norm of its kernel function(s). This concept plays a central role in stability theory, branching representations, scaling limits, and in capturing the endogeneity of the process across applications from high-frequency finance to neuroscience. Hawkes excitation norms, usually concretized as L¹ norms (for scalar kernels) or spectral radii (for matrix-valued multivariate kernels), underpin both operational criteria (e.g., stationarity) and the emergence of distinct macroscopic behaviors in critical and near-critical regimes.

1. Mathematical Definition and Branching Ratio

The Hawkes process intensity at time $t$ is, in its canonical linear form,

$\lambda_t = \mu + \int_0^t \phi(t-s) dN_s$

where $\mu$ is the exogenous (immigrant) intensity and $\phi$ is the excitation kernel. The Hawkes excitation norm is

$\|\phi\|_1 = \int_0^\infty \phi(s) ds$

which is also called the branching ratio. In the multivariate case with $M$ components and kernels $h^{m,m'}$, the excitation is encoded by the $M \times M$ matrix $H = (\|h^{m,m'}\|_1)$, and the system’s excitation norm is then the spectral radius of $H$. For stability,

$$\text{scalar:} \quad \|\phi\|_1 < 1, \qquad \text{multivariate:} \quad \text{spr}(H) < 1$$

is necessary and sufficient for non-explosion (Laub et al., 2015, Leblanc, 21 May 2025).

This norm has probabilistic interpretation: in the cluster (branching) representation, $\|\phi\|_1$ is the mean number of first-generation offspring per event. For mutually exciting models, the norm matrix $H$ quantifies the mean offspring structure (self- and cross-excitation) between types.

2. Role in Stationarity, Stability, and Clustering

Excitation norms govern the phase diagram of Hawkes processes. When $\|\phi\|_1 < 1$, the immigration-birth structure leads to a stationary process: the cluster sizes are finite in expectation, and the process does not explode (Laub et al., 2015, Leblanc, 21 May 2025). If the norm exceeds one, the process admits non-trivial probability of generating infinite points in finite time.

For multivariate and generalized processes with different exciting functions per generation or per coordinate,

$p = \sup_n \|I_n\|_1 < 1$

is sufficient for stationarity, where $I_n$ are the $L^1$ norms (“excitation norms for each generation”) (Mehrdad et al., 2014).

The norm also dictates the effective endogeneity: a realization’s fraction of events directly or indirectly caused by self- or mutual excitation converges (in law) to the excitation norm in the stationary regime.

3. Near Unstable Regime: Scaling Limits and Financial Applications

When the excitation norm approaches the critical value ($\|\phi\|_1 \rightarrow 1^{-}$), the process enters a nearly unstable regime. Consider kernels of the form

$$\phi^{(T)}(t) = a_T \phi(t), \quad a_T = \|\phi^{(T)}\|_1 \to 1$$

and scaling time with $T(1 - a_T) \to \lambda$ as $T \to \infty$. In this regime, the rescaled intensity

$C_t^{(T)} = (1 - a_T)\lambda_{tT}^{(T)}$

converges in law to the integral of a Cox-Ingersoll-Ross process (integrated CIR) (Jaisson et al., 2013): $$X_t = \int_0^t (\mu - X_s)\frac{\lambda}{m}\, ds + \sqrt{\frac{\lambda}{m}}\int_0^t \sqrt{X_s}\, dB_s$$ where $m = \int_0^\infty s \phi(s) ds$.

For price processes modeled as a difference of two nearly unstable Hawkes processes (e.g., buy/sell order flow), this criticality yields Heston-type stochastic volatility limits, naturally reproducing microstructure-induced clustering (high-frequency stylized facts) and macroscopic stochastic volatility (Jaisson et al., 2013).

Thus, the excitation norm’s proximity to $1$ serves as a microstructure parameter that bridges fine-scale clustering (high endogeneity) and emergent stochastic volatility phenomena.

4. Excitation Norms in Generalizations and Multivariate Extensions

Generalizations—including Hawkes processes with different exciting functions, inhibition, and nonlinear or time-varying interactions—retain the centrality of excitation norms, although their definitions may require matrix or recursively constructed norms.

• In the generalized linear case (generation-dependent kernels), the sequence of $L^1$ norms of $I_n$ for each generation determines the overall branching and thus the limiting behavior (Mehrdad et al., 2014).
• In the nonlinear or inhibiting setting, the norm is commonly defined as $\int_0^\infty |\omega_{k,j}(s)| ds$ and stability is ensured if the spectral radius of the (absolute) norm matrix is less than 1 (Chen et al., 2017). This construction guarantees robustness to sign changes and preserves a universal criterion for endogeneity and non-explosion.
• In high dimensions, norms of the form $\int_0^\infty |\omega_{k,j}(s)|ds$ underpin concentration and dependence decay rates, and allow control of network scaling (Chen et al., 2017).

Whether interactions are heterogeneous, inhibitory, or nonlinear, excitation norms provide the analytic substrate for limit theorems, large deviations, and concentration bounds.

5. Impact on Large and Moderate Deviations, Moment Formulas

Excitation norms play a dominant role in asymptotic theory:

• Large deviations principles (LDP): The cumulant generating function of the linear Hawkes process depends recursively on excitation norms; the rate function for deviations in $N_t/t$ is controlled by these quantities (Mehrdad et al., 2014).
• Moderate deviations: Fluctuations between the CLT and LDP scales have quadratic rate functions determined by the collection of excitation norms, i.e., the cumulative series $\sum_n \|I_n\|_1$ (Mehrdad et al., 2014).
• Moment formulas: Exponential moment bounds for counts of points or functionals thereof depend explicitly on the norm and the spectral radius of the interaction matrix; for $N(B)$ the number of points in $B$,

$$\mathbb{E}[\exp(\xi N(B))] < \infty \qquad \text{whenever spr}(H) < 1$$

(Leblanc, 21 May 2025).

Recursive fixed-point equations for transforms (Laplace, Z) also depend on the excitation norm, allowing for tractable characterizations of heavy-tail and scaling limits (Baars et al., 2023).

6. Extensions: Non-Exponential, Delayed, and Variable Memory

Hawkes excitation norms generalize to alternative kernel configurations:

• For non-exponential kernels (e.g., Gamma, Weibull, generalized Pareto), the excitation norm remains $\int_0^\infty \phi(s)\,ds$ but with richer temporal decay and memory properties (Kwan et al., 19 Aug 2024). As the process becomes non-Markovian, the excitation norm still features centrally in consistency and ergodicity results for maximum likelihood inference.
• In delayed Hawkes processes, excitation is effected after a random sojourn. While the L¹ norm of the (randomly shifted) kernel is preserved on average, the temporal spread alters clustering structure and scaling limits—yet the same norm governs moment and scaling bounds (Baars et al., 2023).
• In variable memory or age-dependent models, effective excitation norms may be modulated by state variables (such as “age” since last event), which can suppress the immediate effect of clustering and extend the stability regime beyond what a raw L¹ norm would allow (Raad et al., 2018, Quayle et al., 30 Jul 2025).

In all such models, variants of cumulative $L^1$ norms and their matrix equivalents remain pivotal for predictions of macroscopic behavior, for limit theorems, and for statistical tractability.

7. Applications in Quantitative Finance, Neuroscience, and Insurance

The excitation norm concept is essential for interpreting empirical findings and building models across disciplines:

• High-frequency financial data: Empirical branching ratios close to unity confirm that most market events are endogenously driven, requiring models near instability to capture observed persistence and volatility patterns (Jaisson et al., 2013).
• Neuroscience: The norm determines the extent to which spikes in neuronal networks are endogenously (internally) or exogenously (externally) generated; in variable memory and inhibitory extensions, excitation norms guide inference on network structure and stability (Raad et al., 2018, Quayle et al., 30 Jul 2025, Bonnet et al., 2022).
• Insurance/risk: In risk and ruin modeling, the exponential decay rate of ruin probabilities is explicitly governed by the Laplace exponent, a function of the excitation norms (Mehrdad et al., 2014). In cyber risk, proper separation of endogeneity (kernel norm) and exogenous clustering is of actuarial significance (Boumezoued et al., 2023).

Tabular summary:

Model Type	Excitation Norm Definition	Stationarity/Stability Condition
Scalar linear Hawkes	$\|\phi\|_1$	$\|\phi\|_1 < 1$
Multivariate Hawkes	Spectral radius of $H$	$\text{spr}(H) < 1$
Inhibitory/nonlinear	Spectral radius of $|\omega|$	$\text{spr}(\int |\omega_{k,j}|)$ < 1
Generation-different	$\sup_n \|I_n\|_1$	$p<1$
Delayed/variable memory	Effective $L^1$ norm (modified)	Model-specific, norm-dependent

8. Summary

Hawkes excitation norms, formalized through $L^1$ integrals or spectral radii, act as master parameters governing the microscopic and macroscopic properties of linear and nonlinear, uni- and multivariate, stationary and nonstationary Hawkes models. Their value directly controls stationarity and endogeneity, scaling limits and critical regimes, and analytic tractability for inference and moment estimates. Extensions of the Hawkes framework—allowing for inhibition, stochastic kernels, delays, or variable memory—preserve the centrality of excitation norms, which remain indispensable in rigorous paper and practical application across complex systems.