Optimal parameter estimation for linear SPDEs from multiple measurements

Randolf Altmeyer, Anton Tiepner, Martin Wahl

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Abstract

The coefficients in a second order parabolic linear stochastic partial differential equation (SPDE) are estimated from multiple spatially localised measurements. Assuming that the spatial resolution tends to zero and the number of measurements is nondecreasing, the rate of convergence for each coefficient depends on its differential order and is faster for higher order coefficients. Based on an explicit analysis of the reproducing kernel Hilbert space of a general stochastic evolution equation, a Gaussian lower bound scheme is introduced. As a result, minimax optimality of the rates as well as sufficient and necessary conditions for consistent estimation are established.

OriginalsprogEngelsk
TidsskriftAnnals of Statistics
Vol/bind52
Nummer4
Sider (fra-til)1307-1333
Antal sider27
ISSN0090-5364
DOI
StatusUdgivet - aug. 2024

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