Queue-Reactive Hawkes Models

• Queue-Reactive Hawkes models are point process models that combine exogenous queue-dependent baselines with endogenous self- and mutual-excitation to capture high-frequency event dynamics.
• They employ state-dependent kernels and nonparametric estimation methods to adapt excitation based on real-time queue or limit order book states, improving model parsimonity and empirical fit.
• The models have broad applications in financial microstructure, jump-diffusion option pricing, and queueing theory, offering both analytical clarity and computational efficiency.

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 (Morariu-Patrichi et al., 2018).

• 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 (Wu et al., 2019).

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 (Protter et al., 2021).

• 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 (Arias et al., 2022).

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 (Protter et al., 2021), $\mu_i(Q_{t^-}, t) = M_i(l_i(t)) + \Theta_i(t)$, 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) (Protter et al., 2021).

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 (Morariu-Patrichi et al., 2018, Wu et al., 2019).
• 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 (Protter et al., 2021). 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 (Protter et al., 2021).
• 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 (Koops et al., 2017).

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 (Morariu-Patrichi et al., 2018).

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 (Wu et al., 2019). - 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 (Wu et al., 2019).

• 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 (Protter et al., 2021).
• 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 (Arias et al., 2022).

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 (Morariu-Patrichi et al., 2018).
• 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 (Protter et al., 2021).
• 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 (Morariu-Patrichi et al., 2018).
  • 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 (Koops et al., 2017).

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 (Arias et al., 2022).
• 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 (Koops et al., 2017).

7. Comparative Table of Queue-Reactive Hawkes Models

Model	State-Dependence	Excitation Kernels	Estimation Approach
sdHawkes (Morariu-Patrichi et al., 2018)	Discrete LOB state (spread, imbalance bins)	Exponential, state-specific	MLE, likelihood-splitting
QRH-I, II (Wu et al., 2019)	Queue size (single/multi-queue)	Sum-of-exponentials, nonparam / multiplicative scaling	MLE or least squares
Hawkes+Markovian (Protter et al., 2021)	Bucketized queue + time-of-day	Nonparametric, spline-smoothed	Regularized regression (LASSO), AIC
Queue-Hawkes Jump-Diffusion (Arias et al., 2022)	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.