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

We consider a diffusion process X with values in R(d), whose coefficie nts are smooth enough, and the diffusion coefficient is non-degenerate and depends on an unknown real parameter theta. We are allowed to obs erve the path of X at n times only, and we study here ''random samplin gs'', that is sampling schemes such that the ith sampling time may dep end on the previous i - 1 observations. We prove first the LAMN proper ty as n goes to infinity, for large classes of sequences of such rando m sampling schemes. Second, we exhibit a sequence of random sampling s chemes and associated estimators ($) over cap theta(n) for theta, such that root n(theta(n)-theta) is asymptotically mixed normal, with an a symptotic conditional variance achieving the optimal (over all possibl e random sampling schemes) bound of the LAMN property simultaneously f or all theta.