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.

翻译：本文研究二维系统$\left(Z_t\right)_{t \in[0, T]}=\left(X_t, \lambda_t\right)_{t \in[0, T]}$平稳分布密度$\pi$的非参数估计问题。在该系统中，$X$为Hawkes-扩散过程，$\lambda$表示驱动$X$跳跃的Hawkes过程的随机强度。基于对$(X_t)$在$[0, T]$上一条路径的连续观测，并首先假设$\lambda$已知，我们建立了核估计量$\widehat\pi\left(x^*, y^*\right)$对$\pi\left(x^*,y^*\right)$当$T \rightarrow \infty$时的收敛速率。值得注意的是，该速率受Hawkes强度过程基线参数影响的$y^*$取值所决定。根据$\widehat\pi\left(x^*,y^*\right)$的收敛速率，我们推导出不变密度$\lambda$估计量的收敛速率。随后，我们将研究拓展至$\lambda$未知的情形，通过将$\lambda$的估计量代入核估计量，推导所得估计量的新收敛速率。建立这些收敛速率的证明依赖于具有独立价值的概率论结果。我们引入Girsanov测度变换，将强度为$\lambda$的Hawkes过程转化为具有恒定强度的泊松过程。为此，我们将最初在平稳情形下建立的Hawkes过程指数矩界推广至非平稳情形。最后，我们通过数值研究展示所得估计量的收敛速率。