Grid Based Nonlinear Filtering Revisited: Recursive Estimation & Asymptotic Optimality

This paper revisits grid based recursive approximate filtering of general Markov processes in discrete time, partially observed in conditionally Gaussian noise. The grid based filters considered rely on two types of state quantization, namely, the Markovian type and the marginal type. A set of novel, relaxed sufficient conditions is proposed, ensuring strong and fully characterized pathwise convergence of these filters to the respective MMSE state estimator. In particular, for marginal state quantizations, the notion of conditional regularity of stochastic kernels is introduced which, to the best of the authors' knowledge, constitutes the most relaxed condition under which asymptotic optimality of the respective grid based filters is guaranteed. Further, the convergence results are extended to include filtering of bounded and continuous functionals of the state, as well as recursive approximate state prediction. For both Markovian and marginal quantizations, the whole development of the respective grid-based filters relies more on linear-algebraic techniques and less on measure theoretic arguments, making the presentation considerably shorter and technically simpler.