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

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.