Ergodic properties of the Nonlinear Filter
نویسنده
چکیده
In a recent work [5] various Markov and ergodicity properties of the nonlinear filter, for the classical model of nonlinear filtering, were studied. It was shown that under quite general conditions, when the signal is a Feller-Markov process with values in a complete separable metric space E then the pair process (signal, filter) is also a FellerMarkov process with state space E × P(E), where P(E) is the space of probability measures on E. Furthermore, it was shown that if the signal has a unique invariant measure then, under appropriate conditions, uniqueness of the invariant measure for the above pair process holds within a certain restricted class of invariant measures. In many asymptotic problems concerning approximate filters [6, 7] it is desirable to have the uniqueness of the invariant measure to hold in the class of all invariant measures. In this paper we first show that for a rich class of filtering problems, when the signal has a unique invariant measure, the property of “asymptotic stability” for the filter holds. Using this property of asymptotic stability we then provide sufficient conditions under which the (signal,filter) pair has a unique invariant measure. We also show that, in a certain sense, the property of asymptotic stability is necessary for the uniqueness of the invariant measure.
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