St. Yau et Sst. Yau, Finite dimensional filters with nonlinear drift, XI - Explicit solution ofthe generalized Kolmogorov equation in Brockett-Mitter program, ADV MATH, 140(2), 1998, pp. 156-189
Ever since the technique of the Kalman-Bucy filter was popularized, there h
as been an intense interest in finding new classes of finite-dimensional re
cursive filters. In the late 1970s the concept of the estimation algebra of
a filtering system was introduced. Brockett, Clark, and Mitter proposed to
use the Wei-Norman approach to solve the nonlinear filtering problem. In 1
990, Tam, Wong, and Yau presented a rigorous proof of the Brocket-Mitter pr
ogram which allows one to construct finite-dimensional recursive filters fr
om finite-dimensional estimation algebras. Later Yau wrote down explicitly
a system of ordinary differential equations and generalized Kolmogorov equa
tion to which the robust Duncan-Mortenser-Zakai equation can be reduced. Th
us there remains three fundamental problems in Brockett-Mitter program. The
first is the problem of finding explicit solution to the generalized Kolmo
gorov equation. The second is the problem of finding real-time solution of
a system of ODEs. The third is the Brockett's problem of classification of
finite-dimensional estimation algebras. In this paper, we solve the first p
roblem. (C) 1998 Academic Press.