Nonparametric Bayesian estimation in a multidimensional diffusion model with high frequency data

TL;DR

This paper develops a nonparametric Bayesian approach for estimating multidimensional diffusion models with high-frequency data, achieving optimal convergence rates and establishing theoretical guarantees.

Contribution

It introduces a general posterior contraction rate theorem for such models and demonstrates minimax optimal convergence for Gaussian priors and estimators.

Findings

Posterior and posterior mean converge at minimax optimal rate.

Gaussian priors are effective for high-dimensional diffusion estimation.

Frequentist penalized estimators are also shown to be minimax optimal.

Abstract

We consider nonparametric Bayesian inference in a multidimensional diffusion model with reflecting boundary conditions based on discrete high-frequency observations. We prove a general posterior contraction rate theorem in $L^2$-loss, which is applied to Gaussian priors. The resulting posteriors, as well as their posterior means, are shown to converge to the ground truth at the minimax optimal rate over H\"older smoothness classes in any dimension. Of independent interest and as part of our proofs, we show that certain frequentist penalized least squares estimators are also minimax optimal.