Nonlinear filtering of stochastic dynamical systems
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This paper is a unification of many of the advances in nonlinear filtering made over the last thirty years. After a review of the relevant background material in stochastic processes, we derive the main nonlinear filtering equations for a partially observed system. The system consists of a state process \x t, which is a square integrable semimartingale, and an observation process \y t, which is either a conditional Poisson process or consists of the state process observed in additive white noise. In the other main part of the paper we develop a general theory for approximating the optimal filter of a partially observed diffusion process. We also use computer generated simulations to compare the efficiency of various algorithms for computing these approximations.