  Combining information both within and across trajectories, we propose a
simple estimator for the local regularity of the trajectories of a stochastic
process. Independent trajectories are measured with errors at randomly sampled
time points. Non-asymptotic bounds for the concentration of the estimator are
derived. Given the estimate of the local regularity, we build a nearly optimal
local polynomial smoother from the curves from a new, possibly very large
sample of noisy trajectories. We derive non-asymptotic pointwise risk bounds
uniformly over the new set of curves. Our estimates perform well in
simulations. Real data sets illustrate the effectiveness of the new approaches.
