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 $lambda_t =n qleft(X_tright)$ where $X$ is a continuous It^o semimartingale and $n$ is an integer. Both processes are observed continuously over a fixed period $left[0,Tright]$. 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 Holder class of order $beta$ is $n^{frac{-beta}{2beta+1}}$ 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.