A drift-filtered approach to diffusion estimation for multiscale processes

This paper deals with parameter estimation in the context of so-called multiscale diffusions. The aim is to describe the coarse dynamics of a multiscale system by an approximation in the form of a stochastic differential equation with the coarse (homogenized) drift and diffusion coefficients estimated from numerical simulations of the multiscale system. Recently, it was found that for general stochastic homogenization problems a minimal time interval between consecutive observations must be respected for the estimators to be able to ``see'' the homogenized behavior of the system. If this {\em subsampling} interval is large compared to the coarse dynamics of the system, the standard estimators for these coefficients cannot be used to obtain accurate estimates as their convergence behavior strongly depends on the used subsampling. In this work, we focus on diffusion estimation and propose a procedure based on maximum-likelihood estimation that addresses this problem. First, the contributions due to the drift are filtered from the input data, after which the transformed data is used in the estimation. Due to this preprocessing step, the behavior of the standard estimator changes and it is possible to find good estimates for the homogenized diffusion coefficient as long as the subsampling interval is larger than the minimal required value. Various numerical examples are presented for both known and unknown coarse drift.status: publishe