Parameter Estimation for Stochastic Parabolic Equations: Asymptotic Properties of a Two-Dimensional Projection-Based Estimator

A two-dimensional parameter is estimated from the observations of a random field defined on a compact manifold by a stochastic parabolic equation. Unlike the previous works on the subject, the equation is not necessarily diagonalizable, and no assumptions are made about the eigenfunctions of the operators in the equation. The estimator is based on certain finite-dimensional projections of the observed random field, and the asymptotic properties of the estimator are studied as the dimension of the projection is increased while the observation time is fixed. Simple conditions are found for the consistency and asymptotic normality of the estimator. An application to a problem in oceanography is discussed.