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Filtering for nonlinear systems driven by nonwhite noises: An approximation scheme. (English) Zbl 0786.60058

This paper deals with an approximation approach for the computation of the filter corresponding to a nonlinear model with state and observation equations driven by noise processes having a Markovian representation. For the model in a special form and where the coefficients are supposed to depend causally on the observations, the authors study an approximation scheme based on periodic sampling. In the case when there are only noisy observations, an error bound for the approximation is obtained. In this case, the situation when the system depends on an unknown random parameter is considered, and a result in terms of the probability distribution of this parameter is derived.
The first section deals with the study of various possible situations leading to the considered model among which the practically important case of systems driven by Gaussian processes with stationary derivatives (in the sense of distributions) having rational spectral densities for which it is known that they have Markovian realizations. Now let \(x_ t\) be the unobserved signal at the generic time instant \(t\), and \(u^ t\), \(v^ t\) the trajectories up to time \(t\) of the singular and nonsingular components of the observations, respectively. Then, the following formal relation \[ p(dx_ t \mid u^ t,v^ t)=p(dx_ t,du^ t \mid v^ t)/p(du^ t \mid v^ t) \] between conditional distributions, is motivated by Bayes’ rule. In Sections 2 to 5 firstly it is considered the basic problem of approximating, via time discretization, distributions of the form \(p(z_ t\mid v^ t)\), where \(z_ t\) represents \(x_ t\) and finite samples of \(u^ t\). Then, the approximation of general nonsingular filtering problems is studied. In the last section the nonlinear filtering problem with singular observation noise is considered. It is shown that the periodic sampling method described in the previous sections allows a reasonably good approximate solution to this rather difficult problem.
In conclusion, the paper presents a rather general approximation method also for the singular nonlinear filtering problem with approximation and computation based on the same general scheme. But as the authors observe the computational requirements may be rather heavy, especially in the singular case, where it must to discretize over time the entire trajectory of the singular part of the observations. The paper is very interesting and it is useful to be studied.
Reviewer: G.Orman (Braşov)

MSC:

60G35 Signal detection and filtering (aspects of stochastic processes)
93E11 Filtering in stochastic control theory
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