Gaussian process methods for one-dimensional diffusions: optimal rates and adaptation

TL;DR

This paper investigates Gaussian process-based Bayesian methods for estimating periodic drift functions in one-dimensional diffusions, improving convergence rates and demonstrating adaptive procedures for unknown smoothness levels.

Contribution

It advances the understanding of nonparametric Bayesian estimation for diffusions by improving convergence rates and proposing hierarchical adaptive procedures for Gaussian process priors.

Findings

Enhanced convergence rate results for Gaussian process priors.

Development of hierarchical procedures for adaptation to unknown smoothness.

Demonstration of improved estimation performance in diffusion models.

Abstract

We study the performance of nonparametric Bayes procedures for one-dimensional diffusions with periodic drift. We improve existing convergence rate results for Gaussian process (GP) priors with fixed hyper parameters. Moreover, we exhibit several possibilities to achieve adaptation to smoothness. We achieve this by considering hierarchical procedures that involve either a prior on a multiplicative scaling parameter, or a prior on the regularity parameter of the GP.