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

Abstract

The problem of determining a periodic Lipschitz vector field $b=(b_1,\dots ,b_d)$ 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.

arXiv comment55 pages, to appear in the Annals of Statistics

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• Mixing it up: A general framework for Markovian statistics
• Error Bounds of the Invariant Statistics in Machine Learning of Ergodic Itô Diffusions
• Nonparametric Bayesian estimation in a multidimensional diffusion model with high frequency data