Grid Based Nonlinear Filtering Revisited: Recursive Estimation & Asymptotic Optimality
This work provides incremental improvements in nonlinear filtering theory by simplifying conditions for asymptotic optimality, primarily benefiting researchers in signal processing and control systems.
The paper tackles the problem of grid-based recursive filtering for Markov processes in discrete time with conditionally Gaussian noise, proposing relaxed sufficient conditions that ensure strong pathwise convergence to the minimum mean square error state estimator, with results extended to filtering of functionals and state prediction.
We revisit the development of 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: The \textit{Markovian} type and the \textit{marginal} type. We propose a set of novel, relaxed sufficient conditions, ensuring strong and fully characterized pathwise convergence of these filters to the respective MMSE state estimator. In particular, for marginal state quantizations, we introduce the notion of \textit{conditional regularity of stochastic kernels}, which, to the best of our knowledge, constitutes the most relaxed condition proposed, under which asymptotic optimality of the respective grid based filters is guaranteed. Further, we extend our convergence results, including 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.