Local polynomial estimation of the intensity of a doubly stochastic Poisson process with bandwidth selection procedure

We consider a doubly stochastic Poisson process with stochastic intensity $\lambda_t =n q\left(X_t\right)$ where $X$ is a continuous It\^o semimartingale and $n$ is an integer. Both processes are observed continuously over a fixed period $\left[0,T\right]$. An estimation procedure is proposed in a non parametrical setting for the function $q$ on an interval $I$ where $X$ is sufficiently observed using a local polynomial estimator. A method to select the bandwidth in a non asymptotic framework is proposed, leading to an oracle inequality. If $m$ is the degree of the chosen polynomial, the accuracy of our estimator over the H\"older class of order $\beta$ is $n^{\frac{-\beta}{2\beta+1}}$ if $m \geq \lfloor \beta \rfloor$ and it is optimal in the minimax sense if $m \geq \lfloor \beta \rfloor$. A parametrical test is also proposed to test if $q$ belongs to some parametrical family. Those results are applied to French temperature and electricity spot prices data where we infer the intensity of electricity spot spikes as a function of the temperature.