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New approach and results in nonlinear filtering of continuous-time state space

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posted on 2024-04-03, 17:40 authored by Budhi SuryaBudhi Surya

This paper develops a novel method for solving maximum a posteriori (MAP) and maximum likelihood (ML) nonlinear filtering problems in continuous-time state space. Some distributional identities for the statistics of incomplete information are established as the continuous-time counterpart of those given in \cite{Surya}. Using these identities and the duality principle between estimation theory and optimal control, which was pointed out in \cite{Kalman} for discrete-time state space, a new set of exact explicit filtering equations are derived. They consist of the governing dynamics of MAP and ML state estimators and the corresponding covariance matrix. In particular, unless in the linear state-space, the governing dynamic of covariance matrix is influenced/corrected by the observation process in similar fashion to that of the state estimator. This finding constitutes the main appealing feature of the new nonlinear filtering equations. The results generalize earlier works of \cite{Bucy}, in particular \citep{Bryson,Cox,Jazwinski,Mortensen} for nonlinear state-space. Discrete-time representation of the obtained filtering equation is provided. It serves as an alternative to the extended Kalman-Bucy filter.