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

• 成果类型: Article; Early Access
• 署名作者: Hoffmann, Marc; Ray, Kolyan
• 署名单位: Universite PSL; Universite Paris-Dauphine; Institut Universitaire de France; Imperial College London
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-024-01317-w
• 发表日期: 2024
• 关键词: posterior contraction rates; Inverse problems; drift estimation; differential-equations; ergodic diffusions; convergence-rates; scalar diffusions; inference; inequalities; REGULARITY

摘要

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 L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>2$$\end{document}-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 & ouml;lder 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.