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

TL;DR

This paper develops nonparametric kernel estimators for the stationary density of a Hawkes-diffusion system, analyzing their convergence rates under known and unknown intensity scenarios, with theoretical proofs and numerical validation.

Contribution

It introduces new convergence rate results for kernel estimators of the stationary density in Hawkes-diffusion systems, including cases with unknown intensity, and extends exponential moment bounds to non-stationary processes.

Findings

Convergence rates depend on the Hawkes process parameters.

Extension of exponential moment bounds to non-stationary Hawkes processes.

Numerical results confirm theoretical convergence rates.

Abstract

We investigate the nonparametric estimation problem of the density $\pi$, representing the stationary distribution of a two-dimensional system $\left(Z_t\right)_{t \in[0, T]}=\left(X_t, \lambda_t\right)_{t \in[0, T]}$. In this system, $X$ is a Hawkes-diffusion process, and $\lambda$ 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 $\lambda$ is known, we establish the convergence rate of a kernel estimator $\widehat\pi\left(x^*, y^*\right)$ of $\pi\left(x^*,y^*\right)$ 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 $\widehat\pi\left(x^*,y^*\right)$, we derive the rate of convergence for an estimator of the invariant density $\lambda$. Subsequently, we extend the study to the case where $\lambda$ is unknown, plugging an estimator of $\lambda$ 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 $\lambda$ 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.