---
title: Queue-Reactive Hawkes Models
url: https://www.emergentmind.com/topics/queue-reactive-hawkes-models
type: topic
---

# Queue-Reactive Hawkes Models

Queue-Reactive Hawkes models constitute a class of point process models designed to integrate both the endogenous dynamics of self- and cross-excitation as captured by Hawkes processes, and the exogenous, state-dependent features inherent to queueing or limit order book (LOB) systems. These models are deployed to describe high-frequency market microstructure, general queueing systems under self-exciting input, and self-exciting jump-diffusion frameworks for derivative pricing. The term “queue-reactive” encodes the essential feature: event intensities explicitly depend on the current queue (or book) state, coupling discrete state dynamics and mutually-exciting event flows.

## 1. Mathematical Definitions and Model Classes

Queue-Reactive Hawkes architectures specify multivariate counting processes $(N_1(t),\ldots,N_d(t))$ whose conditional intensities combine state-dependent baseline rates and self-/mutual-excitation depending on both the event history and the present queue state.

Prominent formulations include:

- **State-dependent Hawkes (sdHawkes)**: Introduces a state process $X(t)$ (e.g., queue imbalance bin, bid-ask spread regime) coupled with the event process via transition matrices and queue-dependent kernel parameters. The intensity for event type $e$ is
  $$
  \lambda_e(t)=\nu_e+\sum_{e'=1}^{d_e}\int_{[0,t)} k_{e'\rightarrow e}(t-s, X(s))\,dN_{e'}(s),
  $$
  with the state $X$ updated only at event times via
  $$
  \mathbb{P}[X(T_n) = x' \mid E_n = e', X(T_n^-) = x] = \varphi^{e'}_{x\rightarrow x'}.
  $$
  This defines a Markov chain over states conditional on event arrivals, generalizing both regime-switching Hawkes and continuous-time Markov chains [1809.08060].

- **Queue-reactive Hawkes (QRH-I/QRH-II)**: The single-queue QRH-I variant models event arrival intensities at a fixed price level (e.g., best-bid queue) as
  $$
  \lambda^\ell(t) = \mu^\ell(q(t^-)) + \sum_m \int_0^t \varphi^{\ell m}(t-s)\,dN^m_s,
  $$
  with $q(t)$ the current queue size and $\mu^\ell(\cdot)$ a queue-dependent baseline [1901.08938].

  The QRH-II model jointly evolves all best bid/ask side events, scaling the entire Hawkes excitation by bid/ask queue sizes:
  $$
  \lambda^\ell(t)=f^{(q_b(t),q_a(t))}\left[\mu^\ell + \sum_m \int_0^t \varphi^{\ell m}(t-s)\,dN^m_s\right],
  $$
  with $f^{(q_b,q_a)}$ estimated nonparametrically on a multi-queue grid.

- **Hawkes+Markovian (Order Book Queue Hawkes-Markovian)**: The event intensity is modeled as a sum of a Markovian baseline (explicitly dependent on the discretized queue/LOB state and time-of-day) and nonparametric Hawkes kernels:
  $$
  \lambda_i(t) = \mu_i(Q_{t^-}, t) + \sum_j \int_0^t \varphi_{ji}(t-s)dN_j(s).
  $$
  The baseline $\mu_i(Q_{t^-}, t)$ incorporates both liquidity state (through queue binning) and seasonality [2107.09629].

- **Queue-Hawkes Jump-Diffusion (HQH)**: In the context of option pricing, the queue-reactive Hawkes process provides a jump intensity process $\lambda_t$ that is itself queue-driven:
  $$
  \lambda_t = \lambda^* + \alpha Q_t, \quad d\lambda_t = \alpha (dN_t - dN^Q_t),
  $$
  where each excitation increases $\lambda_t$ discretely, while expiration events remove excitations instantaneously [2205.13321].

## 2. Baseline and Excitation Structure: Parametrization

- **Baseline Intensity**:
  - Can be a function of the current queue size, state variable, or both (e.g., $\mu^\ell(q)$, or $\mu_i(Q_{t^-}, t)$).
  - Captures exogenous influences and slow, state-driven variations in event rates.
  - In [2107.09629], $\mu_i(Q_{t^-}, t) = M_i(l_i(t)) + \Theta_i(t)$, with $M_i$ queue bin (liquidity regime) and $\Theta_i(t)$ capturing intraday seasonality.

- **State-dependent Excitation Kernels**:
  - Exponential or sum-of-exponentials parametrizations are standard (e.g., $k_{e'\rightarrow e}(t, x) = \alpha_{e',x\rightarrow e}\exp(-\beta_{e',x\rightarrow e}t)$ or $\varphi^{\ell m}(u)=\sum_u \alpha^{\ell m}_u \beta_u e^{-\beta_u u}$).
  - Parameters depend on both event types and the queue state at the time of past events.
  - In nonparametric estimation frameworks, kernels $\varphi_{ji}(\cdot)$ are directly estimated as step functions and smoothed (e.g., cubic splines) [2107.09629].

## 3. Estimation Methods and Computational Implementation

- **Maximum Likelihood (MLE)**: Used where kernel and baseline parameters are nonnegative and the log-likelihood is jointly concave. For sdHawkes models, the log-likelihood separates into a state transition term (empirically estimated) and a classical Hawkes term estimated by gradient optimization (e.g., truncated-Newton/C-G) exploiting exponential recursions [1809.08060, 1901.08938].

- **Regularized Regression and Least Squares**: When kernels are high-dimensional or allowed to be nonpositive, $\ell_1$ regularization (LASSO) is used to induce sparsity and prevent overfitting, with coefficients selected by AIC [2107.09629]. Least squares contrast is preferred for nonconvex or signed-kernel variants.

- **Nonparametric and Spline Approximations**: Excitation functions are commonly estimated nonparametrically in discretized time bins, then smoothed by spline interpolation for statistical stability [2107.09629].

- **Markovian and Branching Process Methods for Infinite-Server Queues**: For Hawkes-driven infinite-server queues, both direct PDE/ODE approaches and recursive Poisson-cluster representations are available, providing explicit or numerically stable solutions for transient and steady-state distributions [1707.02196].

## 4. Empirical Results and State-dependent Reflexivity

- **Order Book Microstructure**: All queue-reactive Hawkes models demonstrate that the magnitude and timescale of self- and cross-excitation are state-dependent. For sdHawkes, self-excitation is amplified and cross-excitation decays more slowly in "disequilibrium" states (e.g., spread$>1$, extreme imbalance), with the spectral radius $\rho(x)$ exceeding unity, denoting heightened endogeneity [1809.08060].

  Specific empirical findings include:
  - Baseline rates $\mu^\ell(q)$ are suppressed in large queues for market orders but increase for cancellations (Bund); DAX exhibits weaker dependence [1901.08938].
  - Hawkes (endogenous) fraction of event intensity is 60–80%, varying by state and event type.
  - Queue-reactive baselines enable recovery of heavy-tailed queue distributions missed by pure Markov or Hawkes models and explain state-varying mean reversion in price [1901.08938].

- **Model Comparison Metrics**:
  - Akaike Information Criterion (AIC) consistently selects models incorporating both Hawkes excitations and queue-reactive baselines, with LASSO regularization offering additional improvements in out-of-sample fit [2107.09629].

- **Numerical Complexity**: In jump-diffusion applications, the closed-form characteristic function of a queue-Hawkes process significantly outpaces ODE-based Heston–Hawkes implementations, while retaining the high implied-volatility flexibility associated with self-excitation [2205.13321].

## 5. Extensions, Limitations, and Theoretical Issues

- **Flexibility and Parsimony**: Event-to-state coupling in sdHawkes/QRH models achieves superior parsimony relative to fully extended Hawkes processes with separate intensities per (event,state) pair, reducing the parameter count from $d_e^2 d_x^2$ to $d_e^2 d_x$ while improving fit [1809.08060].

- **Possible Refinements**:
  - Alternative kernel families: power-law or mixture-of-exponentials for heavy tails.
  - State-dependent baseline rates, signed/inhibitory kernels with nonlinearities.
  - Multi-dimensional or joint state spaces (e.g., spread$\times$imbalance).
  - Nonparametric estimation and LASSO regularization to mitigate overfitting in high-dimensional event–state products [2107.09629].

- **Theoretical Challenges**:
  - Stationarity and ergodicity are nontrivial when state-dependent kernels yield $\|k(\cdot,x)\|_1>1$ in some states. This can produce transient sojourns in supercritical regimes; empirical durations are typically brief, but rigorous asymptotic analysis remains open [1809.08060].
  - In infinite-server settings, heavy-traffic and heavy-tailed analysis demonstrate that queue-length distributions inherit self-exciting input properties, with overdispersion and heavy tails whose exponents match the underlying Hawkes process mechanics [1707.02196].

## 6. Applications Beyond Limit Order Books

Queue-reactive Hawkes concepts extend into:

- **Self-exciting jump-diffusion for option pricing**: The HQH framework couples asset price diffusions with queue-reactive Hawkes intensities, enabling efficient European and Bermudan option pricing with explicit Fourier-COS methods and empirically validated implied-volatility flexibility [2205.13321].

- **General queueing theory**: Infinite-server models under Hawkes arrivals reveal the impact of self-excitation on system moments, rare-event tails, and queue-length distributions, thus broadening the theoretical landscape of non-Poisson input queueing systems [1707.02196].

## 7. Comparative Table of Queue-Reactive Hawkes Models

| Model        | State-Dependence      | Excitation Kernels       | Estimation Approach          |
|--------------|----------------------|-------------------------|------------------------------|
| sdHawkes [1809.08060] | Discrete LOB state (spread, imbalance bins) | Exponential, state-specific | MLE, likelihood-splitting    |
| QRH-I, II [1901.08938] | Queue size (single/multi-queue) | Sum-of-exponentials, nonparam / multiplicative scaling | MLE or least squares         |
| Hawkes+Markovian [2107.09629] | Bucketized queue + time-of-day | Nonparametric, spline-smoothed | Regularized regression (LASSO), AIC |
| Queue-Hawkes Jump-Diffusion [2205.13321] | Activation number as queue | Discrete, renewal-style   | Closed-form Fourier methods  |

These architectures collectively provide a rigorous and statistically tractable means to model the coupled evolution of events and queue states. They achieve empirical accuracy, analytic clarity, and flexibility required for high-frequency financial applications, queueing systems, and jump-diffusion modeling in derivative markets.

Source: https://www.emergentmind.com/topics/queue-reactive-hawkes-models