Random sampling in estimation problems for continuous Gaussian processes with independent increments

We study the estimation problem for a continuous (Gaussian) process with independent increments when both the mean (drift) and variance (diffusion coefficient) are functions of the parameter [theta], in the situation where we cannot observe the whole path of the process but we are allowed to sample it at n times only. We are interested in asymptotic properties as the sample size n goes to infinity. Our main result is that there exist random sampling procedures (i.e. the ith sampling time is chosen as a function of the i - 1 previous observations) which are optimal in the sense of maximizing the limit of normalized Fisher information simultaneously for all values of the parameter. Then we construct estimates which are asymptotically normal and with minimal asymptotic estimation variance, again simultaneously for all values of the parameter.