This paper deals with the drift estimation in linear stochastic evolution equations (with emphasis on linear SPDEs) with additive fractional noise (with Hurst index ranging from 0 to 1) via least-squares procedure. Since the least-squares estimator contains stochastic integrals of divergence type, we address the problem of its pathwise (and robust to observation errors) evaluation by comparison with the pathwise integral of Stratonovich type and using its chain-rule property. The resulting pathwise LSE is then defined implicitly as a solution to a non-linear equation. We study its numerical properties (existence and uniqueness of the solution) as well as statistical properties (strong consistency and the speed of its convergence). The asymptotic properties are obtained assuming fixed time horizon and increasing number of the observed Fourier modes (space asymptotics). We also conjecture the asymptotic normality of the pathwise LSE.
The work of PK was supported by the Czech Science Foundation project No. 19-07140S and the work of JŠ was supported by the grant LTAIN19007 Development of Advanced Computational Algorithms for Evaluating Post-surgery Rehabilitation.
We would like to thank to the anonymous referee for valuable comments and suggestions that helped to improve clarity and readability of this paper.
"Pathwise least-squares estimator for linear SPDEs with additive fractional noise." Electron. J. Statist. 16 (1) 1561 - 1594, 2022. https://doi.org/10.1214/22-EJS1990