On Nonlinear Filtering Problems: Structure Theorem and A New Suboptimal Filter

In this thesis, we introduce two methods to solve the nonlinear filtering problem. In chapter 2, we extend Yau and his coauthors' work of Mitter conjecture for low dimensional algebras in nonlinear filtering problem. We prove the Mitter conjecture when the dimension $n=6$. Using this result, we give the structure theorem of six-dimensional estimation algebra. We shall show the structure of six-dimensional estimation algebra is not unique while when $n\leqslant 5$, the structure of estimation algebra is unique. It is hard to solve nonlinear filtering problem when we want to find the optimal filter. In Chapter 3, we first introduce several widely used suboptimal filters, Extended Kalman filter, Unscented Kalman filter, Ensemble Kalman filter, Particle filter, and splitting up method. Then, we introduce a new suboptimal filter for polynomial filtering problems. With our special assumption, we can construct a closed form for conditional mean and conditional moments for the state process. Numerical results show that our new suboptimal filter works perfectly.