Adaptive estimation of intensity in a doubly stochastic Poisson process

In this paper, I consider a doubly stochastic Poisson process with intensity λt=qXt$$ {\lambda}_t=q\left({X}_t\right) $$ where X$$ X $$ is a continuous Itô semi‐martingale. Both processes are observed continuously over a fixed period 0,1$$ \left[0,1\right] $$. I propose a local polynomial estimator for the function q$$ q $$ 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$$ n $$, and q=nq˜$$ q=n\tilde{q} $$, the accuracy of the proposed estimator over the Hölder class of order β$$ \beta $$ is n−β2β+1$$ {n}^{\frac{-\beta }{2\beta +1}} $$ if the degree of the chosen polynomial is greater than ⌊β⌋$$ \left\lfloor \beta \right\rfloor $$ 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.