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

Abstract

The problem of determining a periodic Lipschitz vector field $b=(b_{1},\ldots ,b_{d})$ from an observed trajectory of the solution $(X_{t}:0\le t\le T)$ of the multi-dimensional stochastic differential equation \begin{equation*}dX_{t}=b(X_{t})\,dt+dW_{t},\quad t\geq 0,\end{equation*} where $W_{t}$ 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 $L^{2}$-loss in any dimension, and also for supremum norm loss when $d\le 4$. Further, when $d\le 3$, 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 $\mu _{b}$. The limiting Gaussian process distributions have a covariance structure that is asymptotically optimal from an information-theoretic point of view.