Pathwise least-squares estimator for linear SPDEs with additive fractional noise

Abstract

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.

arXiv comment30 pages, 6 figures

References in corpus (8)

• Estimation of the Hurst parameter from discrete noisy data
• Nonparametric estimation for linear SPDEs from local measurements
• Drift Estimation for Stochastic Reaction-Diffusion Systems
• Diffusivity Estimation for Activator-Inhibitor Models: Theory and Application to Intracellular Dynamics of the Actin Cytoskeleton
• A general drift estimation procedure for stochastic differential equations with additive fractional noise
• Central Limit Theorems and Minimum-Contrast Estimators for Linear Stochastic Evolution Equations
• Pathwise stability of likelihood estimators for diffusions via rough paths
• A space-consistent version of the minimum-contrast estimator for linear stochastic evolution equations