Nonparametric statistical inference for drift vector fields of multi-dimensional diffusions

The problem of determining a periodic Lipschitz vector field b=(b1,.,bd) from an observed trajectory of the solution (Xt:0.t.T) of the multi-dimensional stochastic differential equation dXt=b(Xt)dt+dWt,t.0, where Wt is a standard d-dimensional Brownian motion, is considered. Convergence rates of a penalised least squares estimator, which equals the maximum a posteriori (MAP) estimate corresponding to a high-dimensional Gaussian product prior, are derived. These results are deduced from corresponding contraction rates for the associated posterior distributions. The rates obtained are optimal up to log-factors in L2-loss in any dimension, and also for supremum norm loss when d.4. Further, when d.3, nonparametric Bernstein.von Mises theorems are proved for the posterior distributions of b. From this, we deduce functional central limit theorems for the implied estimators of the invariant measure .b. The limiting Gaussian process distributions have a covariance structure that is asymptotically optimal from an information-theoretic point of view.