Filtering the Maximum Likelihood for Multiscale Problems
arXiv:1305.1918 · doi:10.1137/140952648
Abstract
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.
Keywords: Ergodic filtering, fast mean reversion, homogenization, Zakai equation, maximum likelihood estimation, central limit theory