Pure and Applied Mathematics Quarterly

Volume 10 (2014)

Number 4

On Recent Advance of Nonlinear Filtering Theory: Emphases on Global Approaches

Pages: 685 – 721

DOI: http://dx.doi.org/10.4310/PAMQ.2014.v10.n4.a4


Xue Luo (School of Mathematics and System Sciences, Beihang University, Beijing, China)


The surveys in the field of nonlinear filtering (NLF) are numerous. Most of them are application-oriented and served as the tutorials for the practitioners. The local approaches, including Kalman filter and its invariants, have already been studied from various point of views, due to its off-the-shelf implementation and wide applications. However, it cannot give good estimation of the states in highly nonlinear system or with non-Gaussian initial conditional density functions. Moreover, while the local methods only approximate the mean and variance, the global ones seek the way to directly obtain the conditional density function of the states. Consequently, all the statistical information is acquired. In this survey, we shall briefly go through the local approaches and put emphases on the existing three major global approaches: finite-dimensional NLF, sequential Monte Carlo methods (particle filter) and the Yau-Yau’s on- and off-line solver of Duncan-Mortensen-Zakai’s equation. The discussions are mainly from the mathematical point of view.


nonlinear filtering, global approach, Bayesian framework, Duncan-Mortensen-Zakai’s equation

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