Finite Dimensional Filters with Nonlinear Drift I: A Class of Filters IncludingBoth K alman-Bucy Filters and Benes Filters

摘要

Ever since the technique of the Kalman-Bucy filter was popularized, there hasbeen an intense interest in finding new classes of finite dimensional recursive filters. In the late seventies, the concept of the estimation algebra of a filtering system was introduced. It was proven to be an invaluable tool in the study of nonlinear filtering problems. In 1981, Benes established exact finite dimensional filters for certain diffusions with nonlinear drift. In this paper, we established a simple algebraic necessary and sufficient condition for an estimation algebra of a general class of filtering systems to be finite dimensional. In particular we prove the Mitter conjecture for a large class of nonlinear filtering system. We also give partial solution to the Brockett problem on classification of finite dimensional estimation algebras. Using the Wei-Norman approach, we construct explicitly a general class of finite dimensional recursive filters.