Topics in nonlinear filtering
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In this dissertation, we study the implementation of nonlinear filtering algorithms that can be used in real time applications. In order to implement a filtering algorithm, one has to discretize the state space, the observation space and the time interval. If one discretizes the observation space first, the corresponding equation for the optimal filter is considerably less complicated than in the diffusion case. This is the starting point of our method;First, we focus on the development of a general procedure to solve the filtering problem for Markov semimartingale state processes and jump observation processes. We rewrite the resulting nonlinear equation for the optimal filter into two equations, one describes the evolution of the filter between the observation jump times, the other one updates the filter at the jump times. Then we ignore the nonlinear terms in both equations, and show that the resulting linear equations have at least one weak solution, which is a finite positive measure. It turns out that normalization of this solution yields the optimal nonlinear filter for the problem, which is the unique solution to the filter problem;Second, we consider the discretization of the state space, which leads to filtering equations that are a combination of ordinary differential equations and linear up-dating operations. For this we study the problem of dimension reduction and lower dimensional realizations, in order to reduce the number of variables in the equation, leading to a reduction in computation time. We study exact dimension reduction and approximate dimension reduction. We provide some necessary and sufficient conditions for the problem to have lower dimensional realizations by using three different approaches: invariant linear subspaces, invariant integral submanifolds and exact criteria. We also have developed an efficient and applicable procedure to get approximate dimension reduction and obtain conditions, under which the approximate optimal filter converges to the optimal filter of the problems under consideration. The numerical simulations show that our procedure to solve the nonlinear filtering problem and to reduce the dimension of the filter equations is efficient and applicable.