Nonparametric Bayesian posterior contraction rates for scalar diffusions with high-frequency data

TL;DR

This paper establishes conditions under which Bayesian methods for scalar diffusion models with high-frequency data achieve near-optimal convergence rates for estimating the drift function, adapting to unknown smoothness and sampling regimes.

Contribution

It provides a general theorem for posterior contraction rates in scalar diffusions and identifies natural priors that achieve these rates.

Findings

Bayesian posteriors contract at minimax rates in $L^2$-distance.

Results hold for Besov smoothness classes.

Method adapts to unknown sampling and smoothness.

Abstract

We consider inference in the scalar diffusion model $dX_t=b(X_t)dt+\sigma(X_t)dW_t$ with discrete data $(X_{j\Delta_n})_{0\leq j \leq n}$, $n\to \infty,~\Delta_n\to 0$ and periodic coefficients. For $\sigma$ given, we prove a general theorem detailing conditions under which Bayesian posteriors will contract in $L^2$-distance around the true drift function $b_0$ at the frequentist minimax rate (up to logarithmic factors) over Besov smoothness classes. We exhibit natural nonparametric priors which satisfy our conditions. Our results show that the Bayesian method adapts both to an unknown sampling regime and to unknown smoothness.