Adaptive estimation of intensity in a doubly stochastic Poisson process

In this paper, I consider a doubly stochastic Poisson process with intensity..t = q (Xt) where X is a continuous Ito semi-martingale. Both processes are observed continuously over a fixed period [0, 1]. 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 = nq, the accuracy of the proposed estimator over the Holder class of order.. is n -.. 2..+1 if the degree of the chosen polynomial is greater than.... 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.