April 2024 Consistent inference for diffusions from low frequency measurements
Richard Nickl
Author Affiliations +
Ann. Statist. 52(2): 519-549 (April 2024). DOI: 10.1214/24-AOS2357


Let (Xt) be a reflected diffusion process in a bounded convex domain in Rd, solving the stochastic differential equation


with Wt a d-dimensional Brownian motion. The data X0,XD,,XND consist of discrete measurements and the time interval D between consecutive observations is fixed so that one cannot ‘zoom’ into the observed path of the process. The goal is to infer the diffusivity f and the associated transition operator Pt,f. We prove injectivity theorems and stability inequalities for the maps fPt,fPD,f, t<D. Using these estimates, we establish the statistical consistency of a class of Bayesian algorithms based on Gaussian process priors for the infinite-dimensional parameter f, and show optimality of some of the convergence rates obtained. We discuss an underlying relationship between the degree of ill-posedness of this inverse problem and the ‘hot spots’ conjecture from spectral geometry.


I would like to thank James Norris and Gabriel Paternain for helpful discussions, three anonymous referees and the associate editor for their critical remarks and Matteo Giordano for generating Figures 1–2. The author was supported by EPSRC programme grant EP/V026259.


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Richard Nickl. "Consistent inference for diffusions from low frequency measurements." Ann. Statist. 52 (2) 519 - 549, April 2024. https://doi.org/10.1214/24-AOS2357


Received: 1 November 2022; Revised: 1 January 2024; Published: April 2024
First available in Project Euclid: 9 May 2024

Digital Object Identifier: 10.1214/24-AOS2357

Primary: 35R30 , 62F15 , 62G20

Keywords: Bayesian inverse problems , reflected diffusion process , spectral PCA

Rights: This research was funded, in whole or in part, by EPSRC (UKRI), EP/V026259. A CC BY 4.0 license is applied to this article arising from this submission, in accordance with the grant’s open access conditions.


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Vol.52 • No. 2 • April 2024
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