Nonparametric Bayesian drift estimation for
one-dimensional diffusion processes
We consider diffusions on the circle and establish a Bayesian estimator for the drift function based on observing the local time and using Gaussian priors. Given a standard Girsanov likelihood, we prove that the procedure is well-defined and that the posterior enjoys robustness against small deviations of the local time. A simple method for estimating the local time from high-frequency discrete time observations yielding control of the $L^2$ error is proposed. Complemented by a finite element implementation this enables error-control for a fixed random sample all the way from high-frequency discrete observation to the numerical computation of the posterior mean and covariance. An empirical Bayes procedure is suggested which allows automatic selection of the smoothness of the prior in a given family. Some numerical experiments extend our observations to subsets of the real line other than circles and exhibit more probabilistic convergence properties such as rates of posterior contraction.