Fitting Effective Diffusion Models to Data Associated with a “Glassy" Potential: Estimation, Classical Inference Procedures, and Some Heuristics

A variety of researchers have successfully obtained the parameters of low-dimensional diffusion models using the data that results from atomistic simulations. This raises a variety of questions about efficient estimation, goodness-of-fit tests, and confidence interval estimation. Long time series data and knowledge of the transition density associated with the assumed process model can be of great assistance in obtaining answers to the aforementioned questions. Unfortunately, a closed-form expression for the transition density is rarely available. Estimation and inference are carried out using approximated transition densities (the primary interest is in how Aït-Sahalia's transition density expansions perform in various multiscale contexts). A heuristic strategy for estimating a nonlinear diffusion which does not require one to assume the exact functional form of the diffusion coefficients is also presented. The final part of this study explores how the above findings can be used to help construct a diffusion approximation made possible through a type of homogenization.