Extending the theory of Liptser–Shiryaev filters

Two types of approximate suboptimal Liptser–Shiryaev filters (LSFs) are developed. The first type is based on the generalized Kalman–Bucy filter; the second type, on the parameterization of the a posteriori distribution using the orthogonal expansion method and the quasi-moment method. In principle, the design methods of the LSFs (see below) can be used for obtaining an almost optimal filter with any degree of accuracy. The higher the maximum order of the orthogonal expansion parameters under consideration is, the higher the accuracy of approximation to the optimal estimate will be.