On the under-determined partial differential equation in the nonlinear filtering problems

The Lie algebraic method provides an important research direction for nonlinear filtering theory. By interpreting the Duncan-Mortensen-Zakai equation or its robust form as a partial differential equation with time varying parameters, one derives an approach to filtering that is based on Lie algebra as well as the theory of linear differential operators. The search and construction of the finite dimensional filter are turned into the study of the structure of the estimation algebra. That algebra provides a systematic tool to deal with questions concerning finite dimensional filters. It has led to a number of new results concerning finite dimensional filters . It explains why it is easy to find exact recursive filter for linear dynamical systems but very hard to handle the cubic sensor problem. Some new filters have been discovered. More importantly, the finite dimensionality of the estimation algebra guarantees the explicit construction of the finite dimensional recursive filter, and the filter so constructed is universal. However, as inherited difficulty from the nonlinear filtering, many questions are not satisfiedly answered and still open.