ESTIMATION OF THE DIFFUSION-COEFFICIENT FOR DIFFUSION-PROCESSES - RANDOM SAMPLING

We consider a diffusion process X with values in R(d), whose coefficients are smooth enough, and the diffusion coefficient is non-degenerate and depends on an unknown real parameter theta. We are allowed to observe the path of X at n times only, and we study here ''random samplings'', that is sampling schemes such that the ith sampling time may depend on the previous i - 1 observations. We prove first the LAMN property as n goes to infinity, for large classes of sequences of such random sampling schemes. Second, we exhibit a sequence of random sampling schemes and associated estimators ($) over cap theta(n) for theta, such that root n(theta(n)-theta) is asymptotically mixed normal, with an asymptotic conditional variance achieving the optimal (over all possible random sampling schemes) bound of the LAMN property simultaneously for all theta.