Lie algebraic method in nonlinear filtering with state space dimension up to 6

Abstract

The Kalman-Bucy filter is widely used in modern industry. Despite its usefulness, however, the Kalman-Bucy filter is not perfect. One of its weaknesses is that it needs a Gaussian assumption for the initial data. In this paper we present a Lie algebraic method to solve Duncan-Mortensen-Zakai equation for nonlinear filtering. This method requires only n sufficient statistics, and it allows the initial condition be modeled by an arbitrary distribution. On the other hand, the well-known Makowski method and Haussman-Pardoux method for linear filtering both require a number of sufficient statistics to be a polynomial of degree two in n. However, in the Lie algebraic method, one has to know explicitly the structure of the estimation algebra. In this paper we report the recent results on classification fo finite dimensional maximal rank estimation algebras with state space dimension up to 6.