Diffusion estimation for stochastic multiscale problems

We deal with parameter estimation in the context of so-called multiscale diffusions. For this type of applications, the aim is to describe the coarse dynamics of a multiscale system by an approximation in the form of a stochastic differential equation. The coarse (homogenized) drift and diffusion coefficients therein are estimated from numerical simulations of the multiscale system. 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 \emph{subsampling} interval is large compared to the coarse dynamics of the system, the ``standard'' estimators for drift and diffusion cannot be used to obtain accurate estimates as their convergence behavior strongly depends on the used subsampling. In this talk, we focus on diffusion estimation and propose an estimation 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 accurate estimates for the homogenized diffusion coefficient as long as the minimal subsampling interval is respected.status: publishe