  Given a Wiener process with unknown and unobservable drift, we try to
estimate this drift as effectively but also as quickly as possible, in the
presence of a quadratic penalty for the estimation error and of a fixed,
positive cost per unit of observation time. In a Bayesian framework, where the
unobservable drift is assumed to have a known "prior" distribution, this
question reduces to choosing judiciously a stopping time for an appropriate
diffusion process in natural scale. We establish structural properties of the
solution for the corresponding problem of optimal stopping. In particular, we
show that, regardless of the prior distribution, the continuation region is
monotonically shrinking in time; and provide conditions on the prior
distribution guaranteeing a one-sided stopping region. Finally, we illustrate
the theoretical results through a detailed study of some concrete prior
distributions.
