Abstract

The problem of stochastic filtering is concerned with estimating a signal based upon the partial and noisy observations of the signal. The nonlinear filtering theory has been applied in variety of fields including target detection and tracking, communication networks, mathematical finance, medical sciences, etc. In this thesis, we present some new results on nonlinear filtering with point process observations. These results are motivated by some problems from mathematical finance (cf. Zeng (2003)) and are based upon the novel techniques developed recently by Hu, Ma and Sun (2007). First, we rigorously derive the filtering equations with point process observations under conditions which are weaker than the usual assumptions. Then, we investigate the uniqueness of solutions to the filtering equations, in particular, we obtain the Poisson expansions for the unnormalized optimal filters. Finally, we introduce a recursive numerical method to approximate the unnormalized optimal filters