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June 2020 Nonparametric statistical inference for drift vector fields of multi-dimensional diffusions
Richard Nickl, Kolyan Ray
Ann. Statist. 48(3): 1383-1408 (June 2020). DOI: 10.1214/19-AOS1851


The problem of determining a periodic Lipschitz vector field $b=(b_{1},\ldots ,b_{d})$ from an observed trajectory of the solution $(X_{t}:0\le t\le T)$ of the multi-dimensional stochastic differential equation \begin{equation*}dX_{t}=b(X_{t})\,dt+dW_{t},\quad t\geq 0,\end{equation*} where $W_{t}$ is a standard $d$-dimensional Brownian motion, is considered. Convergence rates of a penalised least squares estimator, which equals the maximum a posteriori (MAP) estimate corresponding to a high-dimensional Gaussian product prior, are derived. These results are deduced from corresponding contraction rates for the associated posterior distributions. The rates obtained are optimal up to log-factors in $L^{2}$-loss in any dimension, and also for supremum norm loss when $d\le 4$. Further, when $d\le 3$, nonparametric Bernstein–von Mises theorems are proved for the posterior distributions of $b$. From this, we deduce functional central limit theorems for the implied estimators of the invariant measure $\mu _{b}$. The limiting Gaussian process distributions have a covariance structure that is asymptotically optimal from an information-theoretic point of view.


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Richard Nickl. Kolyan Ray. "Nonparametric statistical inference for drift vector fields of multi-dimensional diffusions." Ann. Statist. 48 (3) 1383 - 1408, June 2020.


Received: 1 October 2018; Revised: 1 March 2019; Published: June 2020
First available in Project Euclid: 17 July 2020

zbMATH: 07241595
MathSciNet: MR4124327
Digital Object Identifier: 10.1214/19-AOS1851

Primary: 62G20
Secondary: 62F15 , 65N21

Keywords: asymptotics of nonparametric Bayes procedures , Bernstein–von Mises theorem , Penalised least squares estimator , uncertainty quantification

Rights: Copyright © 2020 Institute of Mathematical Statistics


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Vol.48 • No. 3 • June 2020
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