  Filtering and parameter estimation under partial information for multiscale
problems is studied in this paper. After proving mean square convergence of the
nonlinear filter to a filter of reduced dimension, we establish that the
conditional (on the observations) log-likelihood process has a correction term
given by a type of central limit theorem. To achieve this we assume that the
operator of the (hidden) fast process has a discrete spectrum and an
orthonormal basis of eigenfunctions. Based on these results, we then propose to
estimate the unknown parameters of the model based on the limiting
log-likelihood, which is an easier function to optimize because it of reduced
dimension. We also establish consistency and asymptotic normality of the
maximum likelihood estimator based on the reduced log-likelihood. Simulation
results illustrate our theoretical findings.
