Posterior Consistency via Precision Operators for Bayesian Nonparametric Drift Estimation in SDEs

TL;DR

This paper develops a Bayesian method for estimating the periodic drift function of a diffusion process, providing explicit formulas and algorithms, and establishing posterior contraction rates using new local time limit theorems.

Contribution

It introduces a Gaussian prior with a differential precision operator for nonparametric drift estimation in SDEs, with explicit formulas and proven contraction rates.

Findings

Explicit Gaussian posterior formulas derived

Algorithms for implementation provided

Posterior contraction rates established using new local time theorems

Abstract

We study a Bayesian approach to nonparametric estimation of the periodic drift function of a one-dimensional diffusion from continuous-time data. Rewriting the likelihood in terms of local time of the process, and specifying a Gaussian prior with precision operator of differential form, we show that the posterior is also Gaussian with precision operator also of differential form. The resulting expressions are explicit and lead to algorithms which are readily implementable. Using new functional limit theorems for the local time of diffusions on the circle, we bound the rate at which the posterior contracts around the true drift function.