Solution to Brockett´s problem on finite-dimensional estimation algebras of maximal rank in nonlinear filtering

The Kalman-Bucy filter is widely used in modern industry. Despite its usefulness, however, the Kalman-Bucy filter is not perfect. One of the weakness is that it needs a Gaussian assumption for the initial data. The other weakness is that it requires the drift term f(x) be a linear function. Brockett (1981), Brockett and Clark (1980), and Mitter (1979) proposed independently using a Lie algebraic method to solve the Duncan-Mortensen-Zakai equation for nonlinear filtering. This method requires only n sufficient statistics, where n is the state space dimension, and it allows the initial condition to be modeled by an arbitrary distribution. The idea was worked out in detail by Tam, Wong, and Yau (1990) and Yau (1990, 1994). However, in the Lie algebraic method, one has to know explicitly the structure of the estimation algebra. In 1983, Brockett proposed to classify all finite dimensional filters. In this paper, we report more recent results on classification of finite dimensional maximal rank estimation algebras with arbitrary state space dimension