Temporal Marked Hawkes Processes

• Temporal Marked Hawkes Processes are self-exciting point process models that incorporate marks to capture event heterogeneity and cross-event influences.
• They extend classical Hawkes processes through additive or multiplicative mark kernels to model diverse dynamic interactions and temporal clustering.
• Methodologies include likelihood-based estimation, EM algorithms, and modern deep-network adaptations for scalable inference and improved predictive accuracy.

Temporal marked Hawkes processes are self-exciting temporal point processes in which each event is accompanied by a mark from a specified mark space, introducing a rich structure for modeling event heterogeneity, interactions, and excitation patterns. The defining property is that an event’s occurrence increases the future arrival rate, possibly in a mark-dependent and/or cross-mark fashion, for some period of time. These processes are foundational in fields such as seismology, high-frequency finance, insurance, neuroscience, and social network analysis, and underpin a variety of classical, parametric, and modern nonparametric or deep-learning point process models.

1. Mathematical Formulation and Structure

Let $N(t)$ denote a simple point process on $[0,\infty)$ with history $\mathcal H(t)$, and let observed events have associated marks $m_i\in\mathcal M$. The temporal marked Hawkes process defines a joint conditional intensity: $$\lambda(t,m) = \mu(m) + \sum_{t_i < t} g(t - t_i, m, m_i)$$ where:

• $\mu(m)\ge 0$ is the baseline mark-dependent rate,
• $g(\Delta t, m, m') \ge 0$ is the marked triggering kernel,
• $\{ (t_i, m_i) \}$ denotes all past events and their marks.

This model subsumes both the unmarked Hawkes process (when $g$ and $\mu$ are independent of marks) and a wide class of semiparametric models depending on kernel form and mark structure.

Two alternatives for mark impact:

• Additive mark kernels: each mark contributes to the intensity at its own value.
• Multiplicative mark kernels: cross-mark excitation via redistribution kernel $k(m, m')$.

Common choices for the triggering kernel include:

• Marked exponential:

$$g(\Delta t, m, m') = \alpha w(m') e^{-\beta \Delta t} h(m)$$

• Marked power-law (e.g., for aftershock modeling):

$$g(\Delta t, m, m') = k(c + \Delta t)^{-p} k_1(m) k_2(m'),\quad p>1$$

Stationarity requires $$\int_0^\infty \int_{\mathcal M} g(t, m, m')\, \mathrm{d}m\, \mathrm{d}t < 1$$ (Laub et al., 2015, Laub et al., 17 May 2024).

The conditional intensity for arrivals at time $t$ is then $$\lambda^*(t) = \int_{\mathcal M} \lambda(t, m) \, \mathrm{d}m$$, and the conditional mark distribution is $f(m \mid t) = \lambda(t, m)/\lambda^*(t)$.

2. Statistical Inference, Estimation, and Model Validation

Given sequences $\{(t_i, m_i)\}$, parameter estimation is predominantly likelihood-based: $$\ell(\Theta) = \sum_{i=1}^k \log \lambda(t_i, m_i; \Theta) - \int_0^T \int_{\mathcal M} \lambda(s, m; \Theta) \mathrm{d}m\,\mathrm{d}s$$ where $\Theta$ indexes all parameters in $\mu(m)$ and $g(\cdot, \cdot, \cdot)$ (Laub et al., 2015).

For parametric models, direct maximization of the log-likelihood is feasible when $g$ admits closed-form integration; otherwise, recursive or EM-based algorithms are employed. In the EM framework, each event is associated (stochastically) with an immigrant or a unique parent, and the E-step evaluates the posterior probabilities of these assignments: $$p_{0,i} = \frac{\mu(m_i)}{\lambda(t_i, m_i)}, \quad p_{j,i} = \frac{g(t_i - t_j, m_i, m_j)}{\lambda(t_i, m_i)}$$ (Laub et al., 2015).

In multivariate and marked settings, maximum likelihood estimation is supported by asymptotics for the MLE and the Fisher information, and large-sample tests (Wald, likelihood-ratio, and score) enable selection between models of varying complexity while guarding against overfitting (Bonnet et al., 7 Oct 2024).

Model adequacy is routinely checked via the time-change theorem: under the correct model, the integrated intensity at each event time transforms observed times to a unit-rate Poisson process, so QQ-plots or Kolmogorov–Smirnov tests on these transformed intervals assess global fit (Bonnet et al., 7 Oct 2024).

3. Discrete-Time and Compound Extensions

Beyond continuous-time formulations, there exist efficient discrete-time multivariate marked Hawkes processes suitable when observations are binned or naturally slotted. The discrete-time intensity for type $i$ at time $t$ is: $$\lambda_i(t) = \mu_i + \sum_{j=1}^D \sum_{s=1}^{t-1} \phi_{i,j}(t - s, m_j(s)) Y_j(s)$$ with $Y_j(s)$ the count of type-$j$ events at time $s-1$ and marks $m_j(s)$, with excitation kernel $\phi_{i,j}(u, m) = f_{i,j}(m) g_{i,j}(u)$ (Brisley et al., 2023). Efficient $O(N)$ algorithms exploit recurrences when $g_{i,j}$ has a geometric or other recursively computable structure.

In risk and insurance, marked Hawkes risk processes account for both arrival times and jump sizes, with the intensity given by combinations of a (possibly nonlinear) function of past marks and a kernel $h$: $$\lambda_t = \psi \left( \sum_{\tau_i < t} h(t - \tau_i) b(Y_i) \right)$$ and sum claims $R_t = \sum_{\tau_i \le t} Y_i$ (Coutin et al., 10 Sep 2024).

Discrete approximations (Euler-type) to these compound marked Hawkes models retain strong pathwise convergence properties, permitting rigorous error control in Sobolev or Skorokhod metrics, and justifying their use for simulation and estimation on pre-binned data (Coutin et al., 10 Sep 2024).

4. Connections to Branching Processes and Urn Models

Temporal marked Hawkes processes admit a cluster (branching) representation. Each immigrant event initiates a cluster, and offspring events are generated according to the marked kernel, with arbitrary mark-dependent branching structures. This is explicit in the batch-scaling limit of ephemerally self-exciting processes, where entities have finite-lived excitation, and under scaling converge to a general marked Hawkes process: $$\lambda(t) = \mu + \sum_{T_j < t} M_j \bar G(t - T_j)$$ with $M_j$ the mark of the $j$-th immigrant, and $\bar G(u)$ the survivor function of activity durations. The Hawkes kernel thus emerges as an average over latent finite-memory excitation (Daw et al., 2018).

Continuous SE-NBD (self-exciting negative binomial distribution) models, constructed via reinforced Pólya urn processes, further generalize this: their continuous limit yields a marked Hawkes process with gamma-distributed random intensity, capturing both temporal and within-term clustering and introducing flexible heavy-tail behavior at criticality (Hisakado et al., 2021).

5. Modern Extensions: Nonparametric, Deep, and Network Models

Bayesian Nonparametric Inference

Nonparametric marked Hawkes processes allow the marked excitation kernel $h(x, \kappa)$ to be decomposed on a basis—e.g., in time lags and marks—with gamma process priors over basis weights: $$h(x, \kappa) = \sum_{l=1}^L \sum_{m=1}^M \nu_{lm} \, \mathrm{ga}(x|l, \theta^{-1}) \, b_m(\kappa; d)$$ The background intensity $\mu(t)$ itself can be modeled by an Erlang mixture with nonparametric gamma process prior (Kim et al., 27 Nov 2025). This flexible construction enables accurate recovery of magnitude-dependent aftershock productivity and clustering in seismology, outperforming standard ETAS models both in branching-structure recovery and out-of-sample prediction.

Deep Latent and Hypernetwork-Driven Hawkes Models

Recent developments integrate deep state-space architectures with Hawkes dynamics. In the deep linear Hawkes process (DLHP), the intensity is generated via a stack of linear stochastic jump-differential layers, with events of mark $k$ exciting hidden states that are linearly read out to produce per-mark intensities: $$\lambda_t = \exp(C x^{(L)}_{t^-} + b) \in \mathbb{R}^K_+$$ with inter-event evolution given by matrix exponentials and jumps, paralleled across events and marks for computational efficiency (Chang et al., 27 Dec 2024).

The Hyper Hawkes Process (HHP) generalizes further by using latent states $x_t\in\mathbb R^d$ with piecewise-linear dynamics governed by hypernetworks (e.g., GRUs) that produce interval-specific decay operators. The conditional intensity vector is: $\lambda_t = \sigma(\mu + W x_t^-)$ where $\sigma$ is the softplus, and after each event, a mark-specific impulse is added to $x_t$. This structure supports efficient closed-form attribution, inspection of event influence, and state-of-the-art empirical performance on diverse multivariate marked sequence datasets (Boyd et al., 2 Nov 2025).

Network and Community Structure

In temporal networks, events correspond to marked interactions (e.g., directed edges). Parsimonious marked Hawkes models integrate community and influencer structure, associating nodes with communities/hubs/inactives and using blockwise and motif-specific excitation parameters: $$\lambda_{ij}(t) = \mu_{Z_i, Z_j} + \sum_{(k,l)} \int_0^{t} g_{(k,l)\to(i,j)}(t-s) dN_{kl}(s)$$ with $g$ structured by block membership and excitation modes (self, reciprocal, motif excitations). Parameter estimation scales tractably, producing interpretable block, motif, and decay structure (Zhu et al., 29 Jan 2025).

6. Application Domains and Specialized Models

• Seismology: Temporal marked Hawkes processes (notably ETAS) with marks as earthquake magnitudes, where marks influence both intensity and offspring magnitude distributions; magnitude-dependent aftershock models leverage power-law and exponential productivity laws (Laub et al., 2015, Laub et al., 17 May 2024, Kim et al., 27 Nov 2025).
• Finance: Modeling of trades or price jumps with marks as order size or jump magnitude, cross-excitation, and time-varying baselines. Closed-form moment and signature-plot formulae characterize high-frequency microstructure and macroscopic volatility phenomena (e.g., Samuelson effect) (Deschatre et al., 2021).
• Social Systems: Adoption of memes, hashtags, or interactions in social media is naturally modeled through marked Hawkes processes with marks encoding content or participant identity (Laub et al., 2015, Laub et al., 17 May 2024).
• Risk/Insurance & Neuroscience: Compound or marked risk processes where both event rate and amplitude depend on preceding marked events (Coutin et al., 10 Sep 2024).

7. Theoretical Properties and Criticality

• Stationarity: The process is non-explosive iff $$\int_0^\infty \int_{\mathcal M} g(t, m, m')\, d m\, d t < 1$$ (branching ratio less than one).
• Phase Transition: As the effective reproduction/branching ratio approaches unity, marked Hawkes and SE-NBD processes exhibit power-law cluster-size and intensity distributions. The critical exponents and shape depend on the mark law and within-term reinforcement (Hisakado et al., 2021).
• Variance and Heterogeneity: Mark-heterogeneous and SE-NBD processes admit nonzero intensity variance (gamma-distributed intensities), more realistically modeling real-world heterogeneity and overdispersion (missing in classical Hawkes) (Hisakado et al., 2021).

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Temporal marked Hawkes processes synthesize a wide family of self-exciting, history-dependent stochastic models for point data with auxiliary mark structure. They provide interpretable yet flexible frameworks both for classical domains (e.g., aftershock and trade modeling) and for modern applications that require rich event-level heterogeneity, latent structure, and scalable statistical inference. Methodological innovations continue to broaden the inferential scope and computational tractability, from high-dimensional parametric and nonparametric estimation to deep and hypernetwork-based latent modeling (Laub et al., 2015, Hisakado et al., 2021, Laub et al., 17 May 2024, Deschatre et al., 2021, Kim et al., 27 Nov 2025, Chang et al., 27 Dec 2024, Boyd et al., 2 Nov 2025, Zhu et al., 29 Jan 2025).