Efficient and superefficient estimators of filtered Poisson process intensities

Let NK = {NKt, t ∈ [0, T]} be a filtered Poisson process defined on a probability space (Ω,F,(Ft)t∈[0,T],P)$(\Omega,\mathcal {F},(\mathcal {F}_t)_{t\in [0,T]},P)$, and let θ ≔ (θt, t ∈ [0, T]) be a deterministic function which is the intensity of NK under a probability Pθ. In the present paper we prove that the natural maximum likelihood estimator (MLE) NK is an efficient estimator for θ under Pθ. Using Malliavin calculus we construct superefficient estimators of Stein type for θ which dominate, under the usual quadratic risk, the MLE NK. These superefficient estimators are given under the form NtK+DtN˜Klog(F)$N^K_t+D_t^{\widetilde{N}^K}\log (F)$ where F is a random variable satisfying some assumptions and DtN˜K$D_t^{\widetilde{N}^K}$ is the Malliavin derivative with respect to the compensated version N˜K$\widetilde{N}^K$ of NK.