Multivariate Marked Hawkes Processes

• Multivariate marked Hawkes processes are mathematical models that represent event sequences by coupling baseline intensities with history-dependent excitation shaped by additional marks.
• They capture self- and cross-excitation across event types, where marks modulate the impact and timing of previous occurrences.
• Applied in fields like seismology, finance, and social media, these processes offer insights through likelihood-based estimation and stability analysis.

Multivariate marked Hawkes processes are multivariate point processes in which each coordinate has a predictable intensity and each jump carries an additional mark. In a standard marked formulation, the intensity of coordinate $k$ can be written as

$$\lambda_{k,t}=\mu_k\Big(\frac{t}{T}\Big)+\sum_{l=1}^K\int_{[0,t)\times \mathbb R^m}\varphi_{kl}(t-s,x)\,N_l(ds,dx),$$

so elapsed time and mark value jointly determine self- and cross-excitation across coordinates (Lotz, 2024). In the general marked point-process framework, the compensator density factors as $$f^\alpha(t,x,\theta)=\lambda^\alpha(t,\theta)q^\alpha(t,x,\theta)$$, separating event-time intensity from the conditional mark law; for observed events $(T_k,m_k,\kappa_k)$, this yields a full marked likelihood that includes both the intensity term and a mark-density term (Clinet, 2020).

1. Formal structure

A multivariate marked Hawkes process is typically specified on a family of counting measures $\overline N^\alpha(ds,dx)$, $\alpha=1,\dots,d$, where each component records event times together with marks in a space $\mathbb X$ or $E$ (Clinet, 2020). In the exponential-kernel parametric model studied for testing procedures, the observed events are triples $(T_k,m_k,\kappa_k)$, where $m_k\in\{1,\dots,d\}$ is the component label and $$\lambda_{k,t}=\mu_k\Big(\frac{t}{T}\Big)+\sum_{l=1}^K\int_{[0,t)\times \mathbb R^m}\varphi_{kl}(t-s,x)\,N_l(ds,dx),$$0 is the mark, and the $$\lambda_{k,t}=\mu_k\Big(\frac{t}{T}\Big)+\sum_{l=1}^K\int_{[0,t)\times \mathbb R^m}\varphi_{kl}(t-s,x)\,N_l(ds,dx),$$1-th conditional intensity is

$$\lambda_{k,t}=\mu_k\Big(\frac{t}{T}\Big)+\sum_{l=1}^K\int_{[0,t)\times \mathbb R^m}\varphi_{kl}(t-s,x)\,N_l(ds,dx),$$2

Here $$\lambda_{k,t}=\mu_k\Big(\frac{t}{T}\Big)+\sum_{l=1}^K\int_{[0,t)\times \mathbb R^m}\varphi_{kl}(t-s,x)\,N_l(ds,dx),$$3, $$\lambda_{k,t}=\mu_k\Big(\frac{t}{T}\Big)+\sum_{l=1}^K\int_{[0,t)\times \mathbb R^m}\varphi_{kl}(t-s,x)\,N_l(ds,dx),$$4, $$\lambda_{k,t}=\mu_k\Big(\frac{t}{T}\Big)+\sum_{l=1}^K\int_{[0,t)\times \mathbb R^m}\varphi_{kl}(t-s,x)\,N_l(ds,dx),$$5, and the mark enters multiplicatively through $$\lambda_{k,t}=\mu_k\Big(\frac{t}{T}\Big)+\sum_{l=1}^K\int_{[0,t)\times \mathbb R^m}\varphi_{kl}(t-s,x)\,N_l(ds,dx),$$6 (Bonnet et al., 2024).

The general marked likelihood reflects two distinct statistical objects. One part depends on the arrival intensities $$\lambda_{k,t}=\mu_k\Big(\frac{t}{T}\Big)+\sum_{l=1}^K\int_{[0,t)\times \mathbb R^m}\varphi_{kl}(t-s,x)\,N_l(ds,dx),$$7, and the other depends on the conditional mark densities $$\lambda_{k,t}=\mu_k\Big(\frac{t}{T}\Big)+\sum_{l=1}^K\int_{[0,t)\times \mathbb R^m}\varphi_{kl}(t-s,x)\,N_l(ds,dx),$$8. In the quasi-likelihood analysis for multivariate marked point processes, the log-likelihood is decomposed as

$$\lambda_{k,t}=\mu_k\Big(\frac{t}{T}\Big)+\sum_{l=1}^K\int_{[0,t)\times \mathbb R^m}\varphi_{kl}(t-s,x)\,N_l(ds,dx),$$9

where $$f^\alpha(t,x,\theta)=\lambda^\alpha(t,\theta)q^\alpha(t,x,\theta)$$0 is the standard point-process term involving $$f^\alpha(t,x,\theta)=\lambda^\alpha(t,\theta)q^\alpha(t,x,\theta)$$1 and the compensator, while $$f^\alpha(t,x,\theta)=\lambda^\alpha(t,\theta)q^\alpha(t,x,\theta)$$2 is the marked term involving $$f^\alpha(t,x,\theta)=\lambda^\alpha(t,\theta)q^\alpha(t,x,\theta)$$3 integrated against the marked random measure (Clinet, 2020). This decomposition is conceptually important because marked Hawkes models may use marks only as passive labels, may let marks alter excitation amplitudes, or may model the mark process itself.

In graph-theoretic formulations, the integrated kernel amplitudes define the directed interaction structure. With kernels parameterized as $$f^\alpha(t,x,\theta)=\lambda^\alpha(t,\theta)q^\alpha(t,x,\theta)$$4, the weighted adjacency matrix is $$f^\alpha(t,x,\theta)=\lambda^\alpha(t,\theta)q^\alpha(t,x,\theta)$$5, and the support of $$f^\alpha(t,x,\theta)=\lambda^\alpha(t,\theta)q^\alpha(t,x,\theta)$$6 coincides with the Hawkes causal graph: testing $$f^\alpha(t,x,\theta)=\lambda^\alpha(t,\theta)q^\alpha(t,x,\theta)$$7 is interpreted as testing absence of directed causal influence from coordinate $$f^\alpha(t,x,\theta)=\lambda^\alpha(t,\theta)q^\alpha(t,x,\theta)$$8 to coordinate $$f^\alpha(t,x,\theta)=\lambda^\alpha(t,\theta)q^\alpha(t,x,\theta)$$9 (Lotz, 2024).

2. Representations and equivalent constructions

A central structural insight is that marked dependence can sometimes be recoded as multivariate interaction. In “Multivariate Representations of Univariate Marked Hawkes Processes,” the mark space $(T_k,m_k,\kappa_k)$0 is partitioned into measurable bins $(T_k,m_k,\kappa_k)$1, and a univariate marked intensity is represented as

$(T_k,m_k,\kappa_k)$2

Under equicontinuity and continuity assumptions on the target marked intensity, this multivariate unmarked representation approximates a broad non-separable marked Hawkes process in $(T_k,m_k,\kappa_k)$3 arbitrarily well for sufficiently fine partitions (Davis et al., 2024). The construction is asymptotic in $(T_k,m_k,\kappa_k)$4, holds for fixed realizations and finite horizons, and is explicitly not a proof of convergence in Skorokhod topology. It therefore provides an approximation architecture rather than a finite-dimensional exact equivalence in general.

This representation addresses a recurring misconception. It does not develop a fully general theory of multivariate marked Hawkes processes in which each component has its own marks. Rather, it shows that a univariate marked Hawkes process can be approximated by an unmarked multivariate Hawkes process obtained by binning the mark space. In that construction, source mark regions become source components, target mark regions become target components, and mark-dependent offspring effects become cross-excitation parameters $(T_k,m_k,\kappa_k)$5 (Davis et al., 2024).

A different generalization appears in generalized multivariate Hawkes processes. There the multivariate system is treated as a single marked point process whose marks are vectors in

$(T_k,m_k,\kappa_k)$6

so one event time may carry activity in one coordinate, several coordinates simultaneously, and additional within-coordinate marks. This construction allows explicit common event times and simultaneous excitation across coordinates, something classical multivariate Hawkes processes do not model directly (Bielecki et al., 2020). Classical multivariate Hawkes processes arise as a special case when the mark measure is supported only on marks with exactly one active coordinate.

3. Stability, identifiability, and ergodicity

For linear multivariate marked Hawkes processes, stability is typically expressed through an integrated-kernel matrix. In the sparsity-testing framework, the standard subcriticality condition is

$(T_k,m_k,\kappa_k)$7

which reduces to the usual branching-ratio condition under normalized kernels (Lotz, 2024). In the parametric exponential marked model, the paper imposes the normalization $(T_k,m_k,\kappa_k)$8, so stationarity is controlled by the matrix $(T_k,m_k,\kappa_k)$9 through $\overline N^\alpha(ds,dx)$0 (Bonnet et al., 2024).

Approximation-based representations inherit stability only under additional assumptions. For the partition-based multivariate representation of a univariate marked Hawkes process, stationarity of the approximating multivariate process is proved when the target marked Hawkes process is stationary and the productivity $\overline N^\alpha(ds,dx)$1 is deterministic; the stochastic-$\overline N^\alpha(ds,dx)$2 case is explicitly conjectured but not proved (Davis et al., 2024). The same work shows that, for fixed $\overline N^\alpha(ds,dx)$3, the parameters of the multivariate representation are identifiable under assumptions A1–A4, including stationarity, positivity, linear independence of kernels under distinct $\overline N^\alpha(ds,dx)$4, and observation of at least one event of every type (Davis et al., 2024).

In the matrix-exponential marked Hawkes setting, process theory is much stronger. With kernels

$\overline N^\alpha(ds,dx)$5

and component-dependent mark transition kernels $\overline N^\alpha(ds,dx)$6, the augmented state $\overline N^\alpha(ds,dx)$7 is finite-dimensional Markov, where $\overline N^\alpha(ds,dx)$8 stores exponentially decaying excitation coordinates (Clinet, 2020). Under the drift, nondegeneracy, and spectral-type conditions [L1]–[L3] and [ND1]–[ND2], this state process is $\overline N^\alpha(ds,dx)$9-geometrically ergodic. That ergodicity is then the key input for local asymptotic normality of the quasi-log-likelihood and convergence of moments for QMLE and quasi-Bayesian estimators (Clinet, 2020).

Marks also affect identifiability directly. For the multivariate exponential marked Hawkes model with $\alpha=1,\dots,d$0 or $\alpha=1,\dots,d$1, identifiability is proved when, for each relevant interaction, two events with distinct positive marks are observed; the decay $\alpha=1,\dots,d$2 is then identified from exponential relaxation, while two distinct mark values separate $\alpha=1,\dots,d$3 from $\alpha=1,\dots,d$4 (Bonnet et al., 2024).

4. Estimation, testing, and scalable computation

Likelihood-based estimation is the default inferential route for parametric marked Hawkes models. In the exponential marked multivariate model, the full log-likelihood is

$\alpha=1,\dots,d$5

so marks affect inference both through the excitation process and through the mark-density term (Bonnet et al., 2024). In the partition-based multivariate representation, the same paper advocates standard multivariate Hawkes MLE as a practical parameterization method rather than introducing a new optimizer (Davis et al., 2024).

Testing procedures in the parametric literature are largely Wald-, score-, or likelihood-based. For exponential marked Hawkes models, one-coefficient and coefficient-equality tests are built from the Hessian-based covariance estimate $\alpha=1,\dots,d$6, while a score-type test of no mark effect uses the null $\alpha=1,\dots,d$7 and the quadratic statistic

$\alpha=1,\dots,d$8

with asymptotic $\alpha=1,\dots,d$9 behavior under the unmarked null (Bonnet et al., 2024). For multivariate marked Hawkes processes with many independent short trajectories, likelihood-ratio testing under one-sided constraints yields nonstandard chi-bar-square asymptotics because the null places $\mathbb X$0 on the boundary of the parameter space (Lotz, 2024).

Marked models also appear in online and discrete-time estimation. NPOLE-MMHP extends RKHS-based online learning from multivariate Hawkes processes to multivariate marked Hawkes processes by replacing the one-dimensional kernel argument $\mathbb X$1 with the two-dimensional time-mark argument $\mathbb X$2, so the marked functional gradient becomes

$\mathbb X$3

The method does not require the separability assumption $\mathbb X$4, although the paper proves regret and stability only for the unmarked case, not separately for NPOLE-MMHP (Yang et al., 2018).

In discrete time, the marked multivariate Hawkes model for incident monitoring uses conditional Poisson means and a binary alarm mark that modifies excitation amplitude but not decay shape. With geometric kernels, the likelihood and gradient admit recursive evaluation with computational cost linear in the number of observed events, and the marked model outperforms IPP, UHP, and unmarked MHP in predictive log-likelihood on the hospital dataset (Brisley et al., 2023).

Scalability becomes decisive when the “mark” is effectively a large discrete type space. “Learning Multivariate Hawkes Processes at Scale” treats entity identity as a discrete categorical type and develops exact likelihood and gradient computation with runtime $\mathbb X$5, compared with $\mathbb X$6 for standard methods, under sparse-activation assumptions and a low-rank nonnegative factorization of cross-excitation (Nickel et al., 2020). The paper is explicit that this is not a full marked Hawkes model, but it is directly relevant when marks are essentially large discrete type labels.

5. Major model families and adjacent generalizations

The classical linear marked Hawkes family is only one point in a broader landscape. One direction keeps the marked point-process viewpoint but enriches the process dynamics. In compound multivariate Hawkes processes, each event carries random excitation amplitudes $\mathbb X$7 affecting offspring intensities and an additional vector-valued mark $\mathbb X$8 contributing to a compound process

$\mathbb X$9

The resulting multivariate compound marked Hawkes process admits large deviation principles, ruin asymptotics, and asymptotically efficient importance sampling based on an exponential change of measure that preserves the model class (Karim et al., 2022).

A second direction expands the event state itself. The multivariate spatio-temporal neural Hawkes process models events $E$0, where type $E$1 functions as a discrete mark and location $E$2 as a continuous spatial attribute. The model replaces explicit additive kernels by a continuous-time LSTM state and defines type-specific spatial intensities

$E$3

The paper is explicit that this is not a classical marked Hawkes process: there is no additive triggering kernel $E$4, no branching-process interpretation, and excitation or inhibition is implicit in the latent state rather than in directly interpretable kernel coefficients (Chukwuemeka et al., 27 Feb 2026).

A third direction uses structured heterogeneity without explicit marks. In sparsely interacting relational-event Hawkes models with common drivers, the intensity

$E$5

combines local actor-specific parameters $E$6 with global parameters $E$7 and time-varying covariates $E$8. The model is not a genuine marked Hawkes process, but it is highly relevant when actor identity, covariates, and network position play the role of structured mark-like information (Kreiss et al., 4 Apr 2025).

Finally, some nonlinear multivariate Hawkes models move beyond linear mark dependence altogether. Multivariate quadratic Hawkes processes make the intensity depend on quadratic functionals of past signed returns, such as $E$9, and are therefore best understood as second-order, mark-like extensions rather than standard marked Hawkes processes (Aubrun et al., 2022).

6. Applications, limitations, and recurrent misconceptions

Across the literature, marked Hawkes models are used for earthquake aftershock sequences, contagious disease spread, content diffusion on social media platforms, order book dynamics, incident monitoring in a forensic psychiatric hospital, German intraday power prices, terrorism data from Pakistan, and Chicago crime data (Davis et al., 2024). The marks vary correspondingly: spatial location, jump size, alarm indicators, and event magnitudes all appear in concrete models.

Several limitations recur. Continuous marks are often discretized, and the partition-based multivariate representation incurs parameter growth of $(T_k,m_k,\kappa_k)$0; it gains flexibility but loses within-bin resolution and is proved only on finite horizons and fixed realizations (Davis et al., 2024). Parametric exponential marked models provide explicit compensators and practical testing procedures, but they remain tied to multiplicative mark impacts such as $(T_k,m_k,\kappa_k)$1 or $(T_k,m_k,\kappa_k)$2 and to exponential memory kernels (Bonnet et al., 2024). Neural spatio-temporal models allow flexible excitation and inhibition without predefined triggering kernels, yet their interpretability is weaker, and the work explicitly argues that likelihood alone is insufficient for evaluating whether meaningful intensity structure has been learned (Chukwuemeka et al., 27 Feb 2026).

A persistent misconception is that every multivariate Hawkes model with many event types is already a multivariate marked Hawkes process. The recent literature distinguishes carefully among three cases. First, some works are genuinely marked and model $(T_k,m_k,\kappa_k)$3 or $(T_k,m_k,\kappa_k)$4 directly (Clinet, 2020, Lotz, 2024). Second, some works treat type identity as a discrete mark but do not include a separate mark distribution or mark-conditioned kernel family; this is the case for scalable high-dimensional multivariate Hawkes models over entities or channels (Nickel et al., 2020). Third, some works use multivariate structure only as a representation device for a marked process after discretizing the mark space (Davis et al., 2024).

The modern subject is therefore best understood as a family of related constructions rather than a single canonical model. At one end lie linear parametric marked Hawkes processes with explicit kernels, branching ratios, and likelihood theory; at the other lie neural, quadratic, and heavy-tailed multivariate point-process models that retain Hawkes-style history dependence while relaxing classical marked structure. The common thread is that component labels, mark dynamics, and cross-excitation jointly determine how past events reshape the future event landscape.