Nonparametric Bayesian posterior contraction rates for discretely observed scalar diffusions

TL;DR

This paper establishes theoretical results on the rate at which Bayesian posterior distributions concentrate around the true parameters in nonparametric scalar diffusion models with discrete observations, especially in the low-frequency sampling regime.

Contribution

It provides a general theorem giving conditions for prior distributions to achieve minimax optimal contraction rates in nonparametric Bayesian inference for discretely observed diffusions, verified for wavelet series priors.

Findings

Derived new concentration inequalities for empirical processes from discretely observed diffusions.

Proved minimax optimal posterior contraction rates under certain prior conditions.

Validated the theoretical results for natural wavelet series priors.

Abstract

We consider nonparametric Bayesian inference in a reflected diffusion model $dX_t = b (X_t)dt + \sigma(X_t) dW_t,$ with discretely sampled observations $X_0, X_\Delta, \dots, X_{n\Delta}$. We analyse the nonlinear inverse problem corresponding to the `low frequency sampling' regime where $\Delta>0$ is fixed and $n \to \infty$. A general theorem is proved that gives conditions for prior distributions $\Pi$ on the diffusion coefficient $\sigma$ and the drift function $b$ that ensure minimax optimal contraction rates of the posterior distribution over H\"older-Sobolev smoothness classes. These conditions are verified for natural examples of nonparametric random wavelet series priors. For the proofs we derive new concentration inequalities for empirical processes arising from discretely observed diffusions that are of independent interest.