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February 2021 Nonparametric estimation for linear SPDEs from local measurements
Randolf Altmeyer, Markus Reiß
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Ann. Appl. Probab. 31(1): 1-38 (February 2021). DOI: 10.1214/20-AAP1581

Abstract

The coefficient function of the leading differential operator is estimated from observations of a linear stochastic partial differential equation (SPDE). The estimation is based on continuous time observations which are localised in space. For the asymptotic regime with fixed time horizon and with the spatial resolution of the observations tending to zero, we provide rate-optimal estimators and establish scaling limits of the deterministic PDE and of the SPDE on growing domains. The estimators are robust to lower order perturbations of the underlying differential operator and achieve the parametric rate even in the nonparametric setup with a spatially varying coefficient. A numerical example illustrates the main results.

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Randolf Altmeyer. Markus Reiß. "Nonparametric estimation for linear SPDEs from local measurements." Ann. Appl. Probab. 31 (1) 1 - 38, February 2021. https://doi.org/10.1214/20-AAP1581

Information

Received: 1 April 2019; Revised: 1 November 2019; Published: February 2021
First available in Project Euclid: 8 March 2021

Digital Object Identifier: 10.1214/20-AAP1581

Subjects:
Primary: 60F05 , 60H15
Secondary: 35J15 , 62G05

Keywords: Feynman–Kac , Fourth moment theorem , Gaussian process , localised scaling limits , nonparametric estimation

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.31 • No. 1 • February 2021
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