Optimal quantization methods for nonlinear filtering with discrete-time observations. (English) Zbl 1084.62095

Pagès, Gilles; Pham, Huyên

Bernoulli 11, No. 5, 893-932 (2005).

Summary: We develop an optimal quantization approach for numerically solving nonlinear filtering problems associated with discrete-time or continuous-time state processes and discrete-time observations. Two quantization methods are discussed: a marginal quantization and a Markovian quantization of the signal process. The approximate filters are explicitly solved by a finite-dimensional forward procedure. A posteriori error bounds are stated, and we show that the approximate error terms are minimal at some specific grids that may be computed off-line by a stochastic gradient method based on Monte Carlo simulations. Some numerical experiments are carried out: the convergence of the approximate filter as the accuracy of the quantization increases and its stability when the latent process is mixing are emphasized.

MSC:

62M20	Inference from stochastic processes and prediction
60G35	Signal detection and filtering (aspects of stochastic processes)
65C60	Computational problems in statistics (MSC2010)
65C05	Monte Carlo methods
93E11	Filtering in stochastic control theory

Keywords:

Euler scheme; Markov chain; numerical approximation; stationary signal; stochastic gradient descent; vector quantization

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