Open Access
August 2017 Nonparametric Bayesian posterior contraction rates for discretely observed scalar diffusions
Richard Nickl, Jakob Söhl
Ann. Statist. 45(4): 1664-1693 (August 2017). DOI: 10.1214/16-AOS1504


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.


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Richard Nickl. Jakob Söhl. "Nonparametric Bayesian posterior contraction rates for discretely observed scalar diffusions." Ann. Statist. 45 (4) 1664 - 1693, August 2017.


Received: 1 October 2015; Revised: 1 July 2016; Published: August 2017
First available in Project Euclid: 28 June 2017

zbMATH: 06773287
MathSciNet: MR3670192
Digital Object Identifier: 10.1214/16-AOS1504

Primary: 62G05
Secondary: 60J60 , 62F15 , 62G20

Keywords: Bayesian inference , Diffusion model , Nonlinear inverse problem

Rights: Copyright © 2017 Institute of Mathematical Statistics

Vol.45 • No. 4 • August 2017
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