The Annals of Statistics

Nonparametric Bayesian posterior contraction rates for discretely observed scalar diffusions

Richard Nickl and Jakob Söhl

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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 $X_{0},X_{\Delta},\ldots,X_{n\Delta}$. 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 Hölder–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.

Article information

Ann. Statist., Volume 45, Number 4 (2017), 1664-1693.

Received: October 2015
Revised: July 2016
First available in Project Euclid: 28 June 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation
Secondary: 60J60: Diffusion processes [See also 58J65] 62F15: Bayesian inference 62G20: Asymptotic properties

Nonlinear inverse problem Bayesian inference diffusion model


Nickl, Richard; Söhl, Jakob. Nonparametric Bayesian posterior contraction rates for discretely observed scalar diffusions. Ann. Statist. 45 (2017), no. 4, 1664--1693. doi:10.1214/16-AOS1504.

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Supplemental materials

  • The supplement to “Nonparametric Bayesian posterior contraction rates for discretely observed scalar diffusions”. This supplement contains several proofs of results in the main paper and states and proves a proposition on Lipschitz properties of self-adjoint operators.