  Due to their conjugate posteriors, Gaussian process priors are attractive for
estimating the drift of stochastic differential equations with continuous time
observations. However, their performance strongly depends on the choice of the
hyper-parameters. We employ the marginal maximum likelihood estimator to
estimate the scaling and/or smoothness parameter(s) of the prior and show that
the corresponding posterior has optimal rates of convergence. General theorems
do not apply directly to this model as the usual test functions are with
respect to a random Hellinger-type metric. We allow for continuous and
discrete, one- and two-dimensional sets of hyper-parameters, where optimising
over the two-dimensional set of smoothness and scaling hyper-parameters is
shown to be beneficial in terms of the adaptive range.
