In this paper, I consider a doubly stochastic Poisson process with intensity ${\lambda }_t=q\left(X_t\right)$ where $X$ is a continuous Itô semi-martingale. Both processes are observed continuously over a fixed period $\left[0,1\right]$. I propose a local polynomial estimator for the function $q$ on a given interval. Next, I propose a method to select the bandwidth in a nonasymptotic framework that leads to an oracle inequality. Considering the asymptotic $n$, and $q=n\tilde{q}$, the accuracy of the proposed estimator over the Hölder class of order $\beta$ is $n^{\frac{-\beta }{2\beta +1}}$ if the degree of the chosen polynomial is greater than $\lfloor \beta \rfloor$ and it is optimal in the minimax setting. I apply those results to data on French temperature and electricity spot prices from which I infer the intensity of electricity spot spikes as a function of the temperature.

中文翻译：

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双随机泊松过程中强度的自适应估计

在本文中，我考虑具有强度的双随机泊松过程${\lambda }_t=q\left(X_t\right)$在哪里$X$是连续的 Itô 半鞅。在固定时间内连续观察这两个过程$\left[0,1\right]$。我提出了该函数的局部多项式估计器$q$在给定的时间间隔内。接下来，我提出了一种在非渐近框架中选择带宽的方法，该方法会导致预言不等式。考虑渐近$n$， 和$q=n\stackrel{～}{q}$，所提出的估计量在 Hölder 阶次类上的准确性$\beta$是$n^{\frac{-\beta }{2\beta +1}}$如果所选多项式的次数大于$\lfloor \beta \rfloor$并且在极小极大设置下是最佳的。我将这些结果应用于法国气温和电力现货价格的数据，从中我推断出电力现货峰值的强度与温度的函数关系。