Fractional Hawkes Processes

• Fractional Hawkes processes are self-exciting point processes characterized by slowly decaying Mittag-Leffler kernels that produce power-law memory and event clustering.
• Tempered and generalized variants employ inverse subordinators to modulate tail behavior, bridging exponential decay and heavy-tailed dynamics.
• Analytical and simulation techniques, including Laplace transforms and thinning algorithms, facilitate robust estimation and have applications in finance, seismology, and complex systems.

A fractional Hawkes process is a self-exciting point process in which the memory kernel driving the excitation dynamics is governed by a function with slow (power-law or semi-heavy) decay, typically involving Mittag–Leffler or related fractional-calculus structures. This yields event clustering, long-range dependence, and rough-path behavior in sharp contrast to classical Hawkes processes with exponentially decaying kernels. Recent developments include tempered and generalized fractional Hawkes models, incorporating a range of tail-heaviness controls using time changes by inverse subordinators, with important applications in finance, seismology, and other fields where intermediate or rough memory regimes are empirically observed.

1. Mathematical Formulation and Kernel Structure

The canonical univariate fractional Hawkes process $N(t)$ has conditional intensity

$$\lambda(t \mid \mathcal{H}_t) = \lambda_0 + \int_{0}^{t} h(t-s)\,dN(s)$$

where:

• $\lambda_0$ is the baseline intensity,
• $h(\cdot)$ is the excitation kernel,
• $\mathcal{H}_t$ is the process history up to $t$.

Fractional Hawkes models typically specify $h(t) = \alpha\, f_\beta(t)$, where $f_\beta(t)$ is proportional to the density of a positive Mittag–Leffler random variable with index $\beta \in (0,1]$. Explicitly,

$$f_\beta(t) = t^{\beta-1} E_{\beta,\beta}(-t^\beta),$$

with $E_{\beta,\gamma}(\cdot)$ the two-parameter Mittag–Leffler function. The Laplace transform is available in closed form: $\mathcal{L}\{f_\beta\}(s) = \frac{1}{1 + s^\beta}$ for the standard scaling, or $\frac{\gamma}{\gamma + s^\beta}$ for rescaled kernels (Habyarimana et al., 2022, Chen et al., 2020).

Heavy tails are controlled by $\beta$: as $t \to \infty$, $f_\beta(t) \sim t^{-(1+\beta)}$, ensuring slow memory decay, in contrast to the exponential (light-tailed) kernels of the original Hawkes process. These features lead to non-Markovian, over-dispersed event counts and persistent clustering.

2. Subordination and Tempered Fractional Hawkes Processes

The tempered fractional Hawkes process (TFHP) is defined via a stochastic time change: subordinating the base Hawkes process with an inverse tempered stable subordinator (ITSS), which itself incorporates a tempering parameter $\theta>0$ to control eventual exponential cutoff of memory (Gupta et al., 2024). Formally,

$N_{\beta,\theta}(t) = N_H(E_{\beta,\theta}(t)),$

where $E_{\beta,\theta}(t)$ is the right-continuous inverse of a tempered $\beta$-stable subordinator. The intensity becomes $\lambda_H(E_{\beta,\theta}(t))$, and the process interpolates between heavy-tailed and exponentially tempered memory.

Generalization via arbitrary inverse Lévy subordinators (GFHPs) yields processes with fine control over tail behavior, including classical, fractional (power-law), tempered fractional (semi-heavy), and other memory regimes, determined by the Bernstein function $f$ governing the subordinator (Gupta et al., 2024).

Model	Time-Change Structure	Tail Type
Hawkes	None	Exponential (light)
Fractional	$E_\beta(t)$, $\beta$-stable	Power-law (heavy)
Tempered Frac.	$E_{\beta,\theta}(t)$, tempered	Semi-heavy
Generalized	Inverse Lévy subordinator $E_f(t)$	Depends on $f$

3. Analytical Results: Mean, Covariance, Governing Equations

Fractional Hawkes processes possess explicit expressions for several key quantities. The mean intensity $\lambda(t)$ and expected event count $E[N(t)]$ can be derived via Laplace transform methods, exploiting the closed-form for the Mittag–Leffler kernel. For $\beta \in (0,1)$, the mean intensity in standard fractional Hawkes is (Habyarimana et al., 2022)

$$\lambda(t) = \frac{\Lambda_0}{1-\alpha} - \frac{\alpha\Lambda_0}{1-\alpha}E_\beta\big((\alpha-1)\gamma t^\beta\big)$$

and

$$E[N(t)] = \frac{\Lambda_0}{1-\alpha} t - \frac{\alpha \Lambda_0}{1-\alpha} t E_{\beta,2}^1\big((\alpha-1)\gamma t^\beta\big)$$

where $E_\beta$ is the Mittag–Leffler function.

The covariance structure and long-memory property arise via the heavy tails of the kernel, and explicit covariance formulas, often involving generalized Mittag–Leffler functions, are available for the TFHP/GFHP variants (Gupta et al., 2024). Governing difference-differential equations for one-dimensional distributions are provided, involving Caputo-type fractional (and tempered) derivatives with explicit forms for forward and backward Kolmogorov equations: $${}^{C}D_{0+}^{\beta,\theta}\, p_{\beta,\theta}(x,t) = \text{integral-differential terms in } x \text{ and expectations over jump distributions}$$

4. Scaling Limits and Rough Fractional Diffusions

For nearly-unstable, heavy-tailed Hawkes processes, explicit scaling limits connect these models to rough fractional diffusions, notably the fractional Cox–Ingersoll–Ross (fCIR) process (Jaisson et al., 2015, Xu et al., 23 Apr 2025). Under appropriate rescaling (in both subcritical and supercritical regimes), the event-count or integrated intensity converges to a solution of a singular Volterra equation: $$Y_t = F^{\alpha,\lambda}(t) + \frac{1}{\sqrt{\mu^*\lambda}} \int_0^t f^{\alpha,\lambda}(t-s)\sqrt{Y_s}\, dB_s$$ where the kernel is

$$f^{\alpha,\lambda}(t) = \lambda t^{\alpha-1} E_{\alpha,\alpha}(-\lambda t^\alpha)$$

and the corresponding Hurst parameter is $H = \alpha - 1/2$, leading to non-semimartingale, rough process paths when $\alpha \in (1/2,1)$. This regime yields processes with persistent, long-memory volatility roughness, directly linked to observed rough volatility in high-frequency financial data (Jaisson et al., 2015, Xu et al., 23 Apr 2025).

5. Simulation and Inference

Simulating sample paths typically relies on Ogata’s thinning algorithm, with intensity updated using the cumulative Mittag–Leffler kernel across all historical events (Habyarimana et al., 2022, Chen et al., 2020). Closed-form Laplace transforms enable tractable analytical computation and fast numerical inversion of means, aiding Monte Carlo validation and calculation of empirical event count distributions.

Maximum-likelihood estimation is feasible by virtue of explicit likelihood formulas involving only logarithms and Mittag–Leffler function evaluations. Consistency and approximate asymptotic normality have been substantiated via synthetic experiments and regularity arguments under stationarity and ergodicity (Davis et al., 2024).

6. Applications and Extensions

Fractional Hawkes processes have achieved demonstrable utility in seismology—especially for mainshock–aftershock modeling—where the Mittag–Leffler kernel more flexibly captures both the strong burstiness and the long-range memory empirically observed in earthquake clustering, outperforming classical ETAS models in certain regimes (Davis et al., 2024).

In financial applications, particularly in the modeling of limit-order book event flows and volatility dynamics, the agent-based underpinnings and the observed match between heavy-tailed (or tempered) memory in high-frequency data and the predictions from fractional Hawkes scaling limits have provided microstructural justification for “rough volatility” paradigms (Jaisson et al., 2015).

Extensions to multivariate settings, spatio-temporal coupling, and further generalizations via alternative (inverse Lévy) subordinators open broad avenues for modeling and statistical estimation, particularly for phenomena exhibiting intermediate or cross-over memory regimes (Gupta et al., 2024).

7. Open Problems and Future Research

Principal research directions include:

• Statistical estimation of fractional and tempering parameters $(\beta,\theta)$, branching ratios, and scale parameters from observed data.
• Development and analysis of efficient simulation algorithms for events governed by inverse subordinators with non-trivial tail properties.
• Laws of large numbers, functional limit theorems, and large deviations for count processes under GFHP/TFHP constructions.
• Exploration of multivariate, cross-exciting, and spatio-temporal fractional Hawkes models.

The capacity to interpolate between light-tailed, heavy-tailed, and semi-heavy-tailed clustering regimes makes fractional Hawkes processes a powerful framework for studying self-exciting systems with rich memory structures across a range of disciplines (Chen et al., 2020, Habyarimana et al., 2022, Davis et al., 2024, Gupta et al., 2024, Xu et al., 23 Apr 2025, Jaisson et al., 2015).