The exact dynamical equations of optimal nonlinear filtering: New approach and results

This paper develops a novel method for solving nonlinear filtering problems in general continuous-time state-space models. The new method is developed based on an adaptation of the statistical estimation theory for incomplete information which has been primarily developed for maximum likelihood (ML) parameter estimation (Dempster). Distributional identities concerning the score function and information matrix of state are established. Using these identities and the duality principle between ML estimation and optimal control theory, a new set of fully explicit exact recursive filtering equations are derived. They consist of the governing dynamics of the state estimator, the covariance and information matrices. In particular, unless in linear observation, the dynamics of the covariance and information matrices take the form of Riccati equation under the influence of observation process in similar fashion to the dynamics of the state estimator. The results generalize the Kalman-Bucy filter, in particular (Benes,Daum,Kushner,Mortensen,Stratonovich) for nonlinear state-space. A discrete-time representation of the optimal filtering equation is provided for numerical computation. To compare the performance of the optimal filter, a sequential Monte Carlo method (particle filter) is proposed for the time discretization of the state-space. Numerical study confirms the consistency and the accuracy of the exact optimal filter compared to the particle filter's.