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

TL;DR

This paper develops nonparametric Bayesian methods for estimating periodic drift vector fields in multi-dimensional diffusions, achieving near-optimal convergence rates and establishing Bernstein-von Mises theorems for the posterior distributions.

Contribution

It introduces a penalised least squares estimator with proven optimal convergence rates and derives nonparametric Bernstein-von Mises theorems for the posterior in multi-dimensional diffusion models.

Findings

Convergence rates are optimal up to log-factors in $L^2$-loss.

Posterior contraction rates are established and shown to be optimal.

Bernstein-von Mises theorems hold for dimensions up to 3, leading to functional CLTs.

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.