Functional Cramér–Rao bounds and Stein estimators in Sobolev spaces, for Brownian motion and Cox processes

We investigate the problems of drift estimation for a shifted Brownian motion and intensity estimation for a Cox process on a finite interval [0,T], when the risk is given by the energy functional associated to some fractional Sobolev space H01⊂Wα,2⊂L2. In both situations, Cramér–Rao lower bounds are obtained, entailing in particular that no unbiased estimators (not necessarily adapted) with finite risk in H01 exist. By Malliavin calculus techniques, we also study super-efficient Stein type estimators (in the Gaussian case).