Nonparametric estimation of the stationary density for Hawkes-diffusion systems with known and unknown intensity

TL;DR

We study nonparametric estimation of the stationary density $\pi$ of the two-dimensional Hawkes-diffusion system $(X_t, \lambda_t)$, where $X_t$ is driven by a Hawkes jump mechanism with intensity $\lambda_t$. Using kernel density estimators with bandwidths $h_1,h_2$, we derive pointwise convergence rates that depend on the regime of $y^*$ relative to the baseline $\xi$ and whether $\lambda$ is known or unknown. When $\lambda$ is known, the rates separate the contributions of the diffusion and the Hawkes-jump component, yielding faster rates away from $\xi$, and a log-in-T factor in the worst case $y^*=\xi$; with unknown $\lambda$, a plug-in approach yields slower, yet still tractable rates, and a parametric rate for estimating $\lambda$ itself. Central to the analysis are Girsanov-type changes of measure to compare Hawkes-driven processes with Poisson processes, and exponential-moment bounds extended to non-stationary Hawkes dynamics. The results are complemented by numerical experiments illustrating the estimated density's convergence behavior. The work highlights how the hypoelliptic structure and the separation of jump and diffusion influence invariant-density estimation in Hawkes-driven systems.

Abstract

We investigate the nonparametric estimation problem of the density $π$, representing the stationary distribution of a two-dimensional system $\left(Z_t\right)_{t \in[0, T]}=\left(X_t, λ_t\right)_{t \in[0, T]}$. In this system, $X$ is a Hawkes-diffusion process, and $λ$ denotes the stochastic intensity of the Hawkes process driving the jumps of $X$. Based on the continuous observation of a path of $(X_t)$ over $[0, T]$, and initially assuming that $λ$ is known, we establish the convergence rate of a kernel estimator $\widehatπ\left(x^*, y^*\right)$ of $π\left(x^*,y^*\right)$ as $T \rightarrow \infty$. Interestingly, this rate depends on the value of $y^*$ influenced by the baseline parameter of the Hawkes intensity process. From the rate of convergence of $\widehatπ\left(x^*,y^*\right)$, we derive the rate of convergence for an estimator of the invariant density $λ$. Subsequently, we extend the study to the case where $λ$ is unknown, plugging an estimator of $λ$ in the kernel estimator and deducing new rates of convergence for the obtained estimator. The proofs establishing these convergence rates rely on probabilistic results that may hold independent interest. We introduce a Girsanov change of measure to transform the Hawkes process with intensity $λ$ into a Poisson process with constant intensity. To achieve this, we extend a bound for the exponential moments for the Hawkes process, originally established in the stationary case, to the non-stationary case. Lastly, we conduct a numerical study to illustrate the obtained rates of convergence of our estimators.

Figures (2)

• Figure 1: Plot of $\log(h_2) \mapsto \log\widehat{\text{Var}}(\widehat{\pi}_{h_1,h_2}(x^*, y))$. Left graph: for $h_1\in[0.01,0.3]$ with $y= \xi$, right $h_1\in[0.01,0.1]$ with $y=\xi+0.3$.
• Figure 2: Plots of $\log(h_2) \mapsto \log\widehat{\text{Var}}(\widehat{\pi}_{h_2}(y))$. Left graph: for $y= \xi$, right graph for $y=\xi+0.3$.

Theorems & Definitions (30)

• Definition 3.1
• Proposition 3.3
• Proposition 3.4
• Proposition 3.5
• Remark 3.6
• Corollary 3.7
• Proposition 3.8
• Corollary 3.9
• Theorem 4.1
• Proposition 4.2