×

Nonlinear filtering revisited: A spectral approach. (English) Zbl 0873.60030

It is known that the optimal filter \(\widehat f(x(t))\) for \(f(x(t))\) based on observations \(y(s)\), \(s\leq t\), is given by the formula \[ \widehat f(x(t))=\int_{\mathbb{R}^d}f(x)u(t,x)dx\Biggl/ \int_{\mathbb{R}^d}u(t,x)dx, \] where the unnormalized filtering density \(u(t,x)\) is a solution of the Zakai equation. The authors prove the formula \[ u(t,x)=\sum_\alpha{1\over \sqrt{\alpha!}}\varphi_\alpha(t,x)\xi_\alpha(y(\cdot)), \] where \(\xi_\alpha(y(\cdot))\) are Wick polynomials and \(\varphi_\alpha(t,x)\) are deterministic coefficients satisfying a recursive system of Kolmogorov-like equations. The formula separates the observations and parameters of the considered filtering scheme: \(\xi_\alpha(y(\cdot))\) are completely defined by the observation process \(y(t)\) and \(\varphi_\alpha(t,x)\) are determined by coefficients of equations for \(x(t)\), \(y(t)\) and by the initial distribution of the signal process \(x(t)\). The authors develop a numerical approximation scheme for \(u(t,x)\). Some results of numerical simulations are presented.

MSC:

60G35 Signal detection and filtering (aspects of stochastic processes)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
93E11 Filtering in stochastic control theory
93C10 Nonlinear systems in control theory
PDFBibTeX XMLCite
Full Text: DOI