Loglinear Hawkes Processes

• Loglinear Hawkes processes are point process models where the conditional intensity is defined as the exponential of a linear functional of the past, ensuring positivity while allowing for both excitation and inhibition.
• They distinguish themselves from classical linear Hawkes models by incorporating an exponential transfer function that naturally handles negative memory effects and refractory behaviors.
• The model’s analysis focuses on stability, explosion criteria, and Markovian reductions, which are crucial for applications in fields such as neuroscience and finance.

Searching arXiv for recent and foundational papers on loglinear Hawkes processes and closely related nonlinear Hawkes models. {"query":"loglinear Hawkes processes arXiv", "max_results": 10} Loglinear Hawkes processes are a special class of nonlinear Hawkes processes in which the conditional intensity is exponential in a linear functional of the past event history. In the formulation studied most directly in the recent literature,

$$\lambda(t)=\exp\left\{\nu+\int_{(-\infty,t)}h(t-s)\,N(ds)\right\}\mathbf 1\{t<T_\infty\},\qquad t\ge 0,$$

so the log-intensity is affine in the memory term. This specification preserves positivity automatically, allows both excitation and inhibition through the sign of the memory function $h$, and departs sharply from the additive intensity of the classical linear Hawkes model. The principal theoretical developments to date concern explosion, nonexplosion, and stability, especially for kernels with refractory or inhibitory structure that arise in neuroscience (Bielecki et al., 15 Jul 2025).

1. Definition and formal setting

The model is posed for a point process

$$N(\cdot)=\sum_{n\in\mathbb Z}\delta_{T_n}(\cdot)\mathbf 1\{|T_n|<\infty\}$$

with ordered event times

$$\cdots \le T_{-2}\le T_{-1}\le T_0\le 0<T_1\le T_2\le \cdots,\qquad \lim_{n\to-\infty}T_n=-\infty,$$

and explosion time

$T_\infty:=\lim_{n\to\infty}T_n\in(0,\infty].$

If $T_\infty=\infty$ almost surely, the process is nonexplosive; if $T_\infty<\infty$ with positive probability, it is explosive. For a history filtration $\{\mathcal F_t\}$, a nonnegative predictable process $\lambda(t)$ is an intensity if

$$E\!\left[N((a,b])\mid \mathcal F_a\right] = E\!\left[\int_a^b \lambda(t)\,dt\ \Big|\ \mathcal F_a\right],\qquad 0\le a<b.$$

Within this framework, a general nonlinear Hawkes process has intensity

$h$0

and the loglinear Hawkes process is the special case $h$1 (Bielecki et al., 15 Jul 2025).

The model additionally assumes

$h$2

so that the exponent is not $h$3. In the standard nonlinear-Hawkes notation, one may write the memory state as

$h$4

with intensity of the form $h$5; the loglinear case corresponds to an exponential link rather than an affine or merely Lipschitz one (Zhu, 2013).

As in general point-process theory, the compensator

$h$6

and the likelihood

$h$7

remain the basic analytic objects for inference, simulation, and goodness-of-fit diagnostics (Laub et al., 2015).

2. Position within the Hawkes family

The linear Hawkes benchmark uses the additive intensity

$h$8

or, in equivalent event-history form,

$h$9

In that model, each past event contributes an additive decaying term to the current intensity. The loglinear model keeps the same convolutional memory architecture but applies an exponential transfer to the accumulated past instead of adding it directly.

This distinction has two immediate consequences. First, loglinear Hawkes processes allow inhibition naturally: $$N(\cdot)=\sum_{n\in\mathbb Z}\delta_{T_n}(\cdot)\mathbf 1\{|T_n|<\infty\}$$0 may be negative while $$N(\cdot)=\sum_{n\in\mathbb Z}\delta_{T_n}(\cdot)\mathbf 1\{|T_n|<\infty\}$$1 remains positive because positivity is enforced by the exponential map. By contrast, the linear Hawkes specification typically requires $$N(\cdot)=\sum_{n\in\mathbb Z}\delta_{T_n}(\cdot)\mathbf 1\{|T_n|<\infty\}$$2, or essentially nonnegative excitation, to preserve nonnegativity of the intensity. Second, earlier general nonlinear Hawkes results do not directly cover the loglinear case, because the exponential transfer is neither globally Lipschitz nor linearly dominated (Bielecki et al., 15 Jul 2025).

The structural contrast with linear Hawkes is also probabilistic. Linear Hawkes processes admit immigration-birth and population representations in which immigrants arrive exogenously and each event generates offspring according to the fertility kernel. In the age-pyramid formulation,

$$N(\cdot)=\sum_{n\in\mathbb Z}\delta_{T_n}(\cdot)\mathbf 1\{|T_n|<\infty\}$$3

the full past is stored through the ages of prior events (Boumezoued, 2015). That branching viewpoint is central for linear theory, but in the broader nonlinear Hawkes literature the classical branching representation fails once the rate mapping $$N(\cdot)=\sum_{n\in\mathbb Z}\delta_{T_n}(\cdot)\mathbf 1\{|T_n|<\infty\}$$4 is nonlinear; this is one reason loglinear Hawkes processes require separate arguments rather than a direct transfer of linear results (Zhu, 2011).

3. Explosion and nonexplosion

The main direct theorem for loglinear Hawkes processes gives sufficient conditions for both nonexplosion and explosion. Let the process be constructed via thinning with memory function $$N(\cdot)=\sum_{n\in\mathbb Z}\delta_{T_n}(\cdot)\mathbf 1\{|T_n|<\infty\}$$5 and initial condition $$N(\cdot)=\sum_{n\in\mathbb Z}\delta_{T_n}(\cdot)\mathbf 1\{|T_n|<\infty\}$$6. If

$$N(\cdot)=\sum_{n\in\mathbb Z}\delta_{T_n}(\cdot)\mathbf 1\{|T_n|<\infty\}$$7

there exists $$N(\cdot)=\sum_{n\in\mathbb Z}\delta_{T_n}(\cdot)\mathbf 1\{|T_n|<\infty\}$$8 such that

$$N(\cdot)=\sum_{n\in\mathbb Z}\delta_{T_n}(\cdot)\mathbf 1\{|T_n|<\infty\}$$9

and

$$\cdots \le T_{-2}\le T_{-1}\le T_0\le 0<T_1\le T_2\le \cdots,\qquad \lim_{n\to-\infty}T_n=-\infty,$$0

then the process is nonexplosive. The intuitive mechanism is immediate inhibition near lag $$\cdots \le T_{-2}\le T_{-1}\le T_0\le 0<T_1\le T_2\le \cdots,\qquad \lim_{n\to-\infty}T_n=-\infty,$$1: recent events do not create arbitrarily large short-lag amplification, so the number of points on each interval $$\cdots \le T_{-2}\le T_{-1}\le T_0\le 0<T_1\le T_2\le \cdots,\qquad \lim_{n\to-\infty}T_n=-\infty,$$2 can be dominated by a Poisson count with finite mean (Bielecki et al., 15 Jul 2025).

The complementary sufficient condition for explosion is short-lag positivity. If there exists $$\cdots \le T_{-2}\le T_{-1}\le T_0\le 0<T_1\le T_2\le \cdots,\qquad \lim_{n\to-\infty}T_n=-\infty,$$3 such that

$$\cdots \le T_{-2}\le T_{-1}\le T_0\le 0<T_1\le T_2\le \cdots,\qquad \lim_{n\to-\infty}T_n=-\infty,$$4

and

$$\cdots \le T_{-2}\le T_{-1}\le T_0\le 0<T_1\le T_2\le \cdots,\qquad \lim_{n\to-\infty}T_n=-\infty,$$5

then the process is explosive, in the sense that

$$\cdots \le T_{-2}\le T_{-1}\le T_0\le 0<T_1\le T_2\le \cdots,\qquad \lim_{n\to-\infty}T_n=-\infty,$$6

The proof works with interarrival times $$\cdots \le T_{-2}\le T_{-1}\le T_0\le 0<T_1\le T_2\le \cdots,\qquad \lim_{n\to-\infty}T_n=-\infty,$$7 and shows that the event of infinitely many arrivals in $$\cdots \le T_{-2}\le T_{-1}\le T_0\le 0<T_1\le T_2\le \cdots,\qquad \lim_{n\to-\infty}T_n=-\infty,$$8 has positive probability through an infinite product lower bound. With

$$\cdots \le T_{-2}\le T_{-1}\le T_0\le 0<T_1\le T_2\le \cdots,\qquad \lim_{n\to-\infty}T_n=-\infty,$$9

one obtains

$T_\infty:=\lim_{n\to\infty}T_n\in(0,\infty].$0

and the product is strictly positive because the corresponding series converges (Bielecki et al., 15 Jul 2025).

The theorem covers several practically relevant classes of memory functions. The cited examples include the following.

• Finite initial mass: if $T_\infty:=\lim_{n\to\infty}T_n\in(0,\infty].$1, both initial-condition assumptions are automatic.
• Eventually nonpositive $T_\infty:=\lim_{n\to\infty}T_n\in(0,\infty].$2: if $T_\infty:=\lim_{n\to\infty}T_n\in(0,\infty].$3 is bounded above and eventually nonpositive, the nonexplosion condition holds for any initial condition.
• Eventually nonnegative $T_\infty:=\lim_{n\to\infty}T_n\in(0,\infty].$4: if $T_\infty:=\lim_{n\to\infty}T_n\in(0,\infty].$5 is bounded below and eventually nonnegative, the explosion condition holds for any initial condition.
• Controlled initial intensity from above: if $T_\infty:=\lim_{n\to\infty}T_n\in(0,\infty].$6 and $T_\infty:=\lim_{n\to\infty}T_n\in(0,\infty].$7 is integrable decreasing, the nonexplosion condition holds.
• Controlled initial intensity from below: if $T_\infty:=\lim_{n\to\infty}T_n\in(0,\infty].$8 and $T_\infty:=\lim_{n\to\infty}T_n\in(0,\infty].$9, the explosion condition holds.

These criteria formalize the short-lag dichotomy emphasized in the paper: nonpositive behavior near zero acts as a refractory mechanism, whereas strictly positive behavior near zero can produce arbitrarily rapid cascades (Bielecki et al., 15 Jul 2025).

4. Stability and stationary behavior

The stability notion used for loglinear Hawkes processes is stability in distribution with respect to an initial condition $T_\infty=\infty$0. This means that there exists a stationary version $T_\infty=\infty$1 on all of $T_\infty=\infty$2, and that the shifted process

$T_\infty=\infty$3

converges weakly to $T_\infty=\infty$4 as $T_\infty=\infty$5. An equivalent formulation is that for every continuous compactly supported $T_\infty=\infty$6,

$T_\infty=\infty$7

A sufficient stability theorem is available in the inhibitory regime. If $T_\infty=\infty$8 is nonpositive and

$T_\infty=\infty$9

and if the initial condition satisfies

$T_\infty<\infty$0

then the loglinear Hawkes dynamics are stable in distribution with respect to $T_\infty<\infty$1. Under the same conditions there is only one stationary process following the dynamics on $T_\infty<\infty$2 (Bielecki et al., 15 Jul 2025).

The argument exploits a reduction to bounded-Lipschitz nonlinear Hawkes theory. When $T_\infty<\infty$3, one may define

$T_\infty<\infty$4

which is bounded and Lipschitz with Lipschitz constant $T_\infty<\infty$5. Because the actual memory term remains in the $T_\infty<\infty$6 regime, the original loglinear intensity coincides with this bounded-Lipschitz version, allowing the use of Brémaud–Massoulié-type stability results for nonlinear Hawkes processes (Bielecki et al., 15 Jul 2025).

The same paper also derives a necessary condition for stability. If the dynamics are stable and the stationary limit has constant mean intensity

$T_\infty<\infty$7

then necessarily

$T_\infty<\infty$8

Since the right-hand side is maximized at $T_\infty<\infty$9, one obtains the universal bound

$\{\mathcal F_t\}$0

If $\{\mathcal F_t\}$1, the stationary process must also be nontrivial: $\{\mathcal F_t\}$2

The qualitative message is that negative memory favors stability, while positive short-lag memory favors explosion. The cited work emphasizes kernels seen in neuroscience, including a refractory dip directly after a spike, possibly followed by an overshoot and then a decaying tail. It also suggests that stable mixed excitation-inhibition models will require sharper conditions than those currently available (Bielecki et al., 15 Jul 2025).

5. Markovian reductions and asymptotic theory in the broader nonlinear literature

Loglinear Hawkes processes lie inside the general nonlinear family

$\{\mathcal F_t\}$3

so they inherit the standard Markovian reductions available for special kernels. If

$\{\mathcal F_t\}$4

then

$\{\mathcal F_t\}$5

satisfies

$\{\mathcal F_t\}$6

For sums of exponentials,

$\{\mathcal F_t\}$7

the coordinate processes

$\{\mathcal F_t\}$8

form a finite-dimensional Markov process with generator

$\{\mathcal F_t\}$9

In the loglinear specialization, the rate mapping is $\lambda(t)$0, so exponential and sum-of-exponentials kernels provide finite-dimensional state representations (Zhu, 2011).

These Markovian reductions are the basis of a substantial asymptotic theory for nonlinear Hawkes processes more generally. For Markovian nonlinear Hawkes processes with suitable sublinear rate assumptions, one has

$\lambda(t)$1

yielding a level-1 large deviation principle for $\lambda(t)$2 with speed $\lambda(t)$3 (Zhu, 2011). Under decreasing $\lambda(t)$4, finite first moment $\lambda(t)$5, positive increasing $\lambda(t)$6-Lipschitz $\lambda(t)$7, and $\lambda(t)$8, the broader nonlinear-Hawkes theory also gives a functional CLT and process-level large deviations (Zhu, 2013).

Those results delineate the technical distinctiveness of the loglinear model. The exponential transfer falls outside the globally Lipschitz and linear-growth regimes that dominate earlier nonlinear Hawkes theory, so the available loglinear results focus instead on explosion, nonexplosion, and stability. A plausible implication is that asymptotic results for loglinear Hawkes processes will need to combine the Markovian state reductions above with arguments tailored to exponential growth, rather than relying directly on the sublinear or Lipschitz frameworks (Zhu, 2013).

6. Adjacent models, recurring confusions, and methodological boundaries

Several nearby models are sometimes conflated with loglinear Hawkes processes because they contain an exponential factor or admit a log-intensity decomposition. The cited literature distinguishes them explicitly.

Model	Defining structure	Relation to loglinear Hawkes
Loglinear Hawkes	$\lambda(t)$9	Log-intensity affine in past events
msdHawkes	$$E\!\left[N((a,b])\mid \mathcal F_a\right] = E\!\left[\int_a^b \lambda(t)\,dt\ \Big|\ \mathcal F_a\right],\qquad 0\le a<b.$$0	Multiplicative state modulation of a Hawkes intensity, not the canonical loglinear form
Shallow Neural Hawkes	$$E\!\left[N((a,b])\mid \mathcal F_a\right] = E\!\left[\int_a^b \lambda(t)\,dt\ \Big|\ \mathcal F_a\right],\qquad 0\le a<b.$$1, with $$E\!\left[N((a,b])\mid \mathcal F_a\right] = E\!\left[\int_a^b \lambda(t)\,dt\ \Big|\ \mathcal F_a\right],\qquad 0\le a<b.$$2	Neural non-parametric kernel estimation for a standard Hawkes intensity, not intensity-level loglinearity
Hawkes with different exciting functions	$$E\!\left[N((a,b])\mid \mathcal F_a\right] = E\!\left[\int_a^b \lambda(t)\,dt\ \Big|\ \mathcal F_a\right],\qquad 0\le a<b.$$3, $$E\!\left[N((a,b])\mid \mathcal F_a\right] = E\!\left[\int_a^b \lambda(t)\,dt\ \Big|\ \mathcal F_a\right],\qquad 0\le a<b.$$4	Generalized linear cluster model with generation-dependent kernels

The multiplicative state-dependent model for limit-order-book dynamics,

$$E\!\left[N((a,b])\mid \mathcal F_a\right] = E\!\left[\int_a^b \lambda(t)\,dt\ \Big|\ \mathcal F_a\right],\qquad 0\le a<b.$$5

has

$$E\!\left[N((a,b])\mid \mathcal F_a\right] = E\!\left[\int_a^b \lambda(t)\,dt\ \Big|\ \mathcal F_a\right],\qquad 0\le a<b.$$6

but the literature is explicit that this is not a canonical loglinear Hawkes model because the exponential factor modulates an existing Hawkes intensity rather than acting as the transfer applied to a baseline-plus-excitation predictor (Sfendourakis et al., 2021). Likewise, "Shallow Neural Hawkes" parameterizes each excitation kernel as the exponential of a one-hidden-layer ReLU network, but the intensity remains the standard additive multivariate Hawkes intensity; the log/exp structure is used only to enforce kernel positivity, not to define a loglinear intensity model (Joseph et al., 2020). The generalized Hawkes process with different exciting functions is again linear in its generation-wise cluster construction, although it is methodologically relevant for branching intuition, multivariate embeddings, and large-deviation calculations (Mehrdad et al., 2014).

Methodological boundaries are equally sharp on the inferential side. The adjacent literature includes direct likelihood maximization and EM-type estimation for multiplicatively state-modulated Hawkes models, as well as stochastic-gradient-compatible likelihood decompositions for neural non-parametric kernel estimation (Sfendourakis et al., 2021). High-dimensional Bayesian posterior contraction results are available for sparse linear Hawkes processes, including control of empirical $$E\!\left[N((a,b])\mid \mathcal F_a\right] = E\!\left[\int_a^b \lambda(t)\,dt\ \Big|\ \mathcal F_a\right],\qquad 0\le a<b.$$7 intensity loss and direct parameter $$E\!\left[N((a,b])\mid \mathcal F_a\right] = E\!\left[\int_a^b \lambda(t)\,dt\ \Big|\ \mathcal F_a\right],\qquad 0\le a<b.$$8 loss, but that work states explicitly that its methods do not directly handle loglinear or more general nonlinear Hawkes models (Rousseau et al., 28 Oct 2025). The current direct theory for loglinear Hawkes processes is therefore primarily foundational rather than inferential: it establishes when the model exists, when it explodes, and when inhibitory dynamics converge to a unique stationary law.