Hawkes process with a diffusion-driven baseline: long-run behavior, inference, statistical tests

TL;DR

This work extends Hawkes processes by coupling them with a diffusion-driven stochastic baseline, enabling the spontaneous event rate to adapt to continuously evolving covariates. The authors establish strong probabilistic foundations, including exponential ergodicity of the joint diffusion–memory process, LLN/CLT for event counts, and mixing properties under varying covariate assumptions. They develop a comprehensive inference framework based on maximum likelihood, proving consistency, asymptotic normality, and, under stronger mixing, moment convergence, while also introducing hypothesis tests to assess the covariate’s relevance and model adequacy. Theoretical results are complemented by extensive simulations, including OU-driven and hypoelliptic diffusion setups, to illustrate estimation, testing procedures, and goodness-of-fit corrections in practice.

Abstract

Event-driven systems in fields such as neuroscience, social networks, and finance often exhibit dynamics influenced by continuously evolving external covariates. Motivated by these applications, we introduce a new class of multivariate Hawkes processes, in which the spontaneous rate of events is modulated by a diffusion process. This framework allows the point process to adapt dynamically to continuously evolving covariates, capturing both intrinsic self-excitation and external influences. In this article, we establish the probabilistic properties of the coupled process, proving stability and ergodicity under moderate assumptions. Classical functional results, including law of large numbers and mixing properties, are extended to this diffusion-driven setting. Building on these results, we study parametric inference for the Hawkes component: we derive consistency and asymptotic normality of the maximum likelihood estimator in the long-time regime, and derive stronger convergence results under additional assumptions on the covariate process. We further propose hypothesis testing procedures to assess the statistical relevance of the covariate. Simulation studies illustrate the validity of the asymptotic results and the effectiveness of the proposed inference methods. Overall, this work provides theoretical and practical foundations for diffusion-driven Hawkes models.

Figures (3)

• Figure 1: Trajectory of the covariate process $X$, with colored dots indicating event times —yellow for $N_1$, green for $N_2$— along the path (Figure \ref{['fig:simu1Process']}), together with the associated intensity functions (Figure \ref{['fig:simu1Intensity']}). Figure \ref{['fig:simu1Stat']} shows the distribution of the test statistic for $\mathrm{H_0}: \mu^{\star}_{i,1} = \mu^{\star}_{i,2}$ with $i \in \{1,2\}$. For Figure \ref{['fig:simu1Process']}--\ref{['fig:simu1Intensity']}$T= 10$ whereas for Figure \ref{['fig:simu1Stat']}$T=3000$ and $n=300$ repetitions were used.
• Figure 2: Left: trajectory of the process $X$ over time, with green dots indicating the event times for $N_1$ along the path. Right: histograms of the estimators under the null hypotheses $\mathrm{H_0}: \mu_1^\star = 0.2$ (top) and $\mathrm{H_0}: \mu_2^\star = 0.5$ (bottom). For this simulation $T= 2000$ and $n=300$ repetitions were used.
• Figure 3: Illustration of Algorithm \ref{['algo:GofCorr']}. Green points show the distribution of p-values when the corrected increments $(\widehat{E}_i(T))$, defined in Equation \ref{['eq:Eicorrected']}, are tested directly against a unit exponential distribution using a KS test. Red points correspond to p-values obtained after randomly subsampling a subset of size $N(T)^{2/3}$ from $(\widehat{E}_i(T))$ and applying the KS test to this subset. The p-values are compared to a standard uniform distribution using a QQ plot: the theoretical quantiles of the uniform distribution are shown as a black line, with 95% confidence intervals represented as the blue region.

Theorems & Definitions (12)

• Theorem 1: Existence
• proof
• Remark 2: General Markovian conditions for $X$
• Theorem 3
• Proposition 1
• Theorem 4
• Remark 5
• Theorem 6: Mixing Properties of the Intensity
• Theorem 7: Asymptotic normality of the MLE
• Remark 8