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Exponential stability for nonlinear filtering. (English) Zbl 0888.93057

The authors study the asymptotic dependence of the optimal filter on the initial law of the signal, and the relation between this dependence and the corrupting noise intensity. This is relevant for numerical and practical computations of the approximations of the filter. The technique is based on considering the unnormalized filtering process as a positive flow in the space of measures, acting on the initial measure and on bounding the amount of contraction generated by the flow, at a discrete skeleton of time instances. The Hilbert metric and the related Birkhoff contraction coefficient are used.
Some examples are treated: the exponential convergence rate is proved; explicit bounds on the rate are provided; one considers the following cases: discrete time filtering, filtering of a diffusion on a compact manifold, discrete time countable state space process.

MSC:

93E11 Filtering in stochastic control theory
60J57 Multiplicative functionals and Markov processes
93C10 Nonlinear systems in control theory
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