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Asymptotic stability of the Wonham filter: ergodic and nonergodic signals
 SIAM J. Control Optim
"... Abstract. Stability problem of the Wonham filter with respect to initial conditions is addressed. The case of ergodic signals is revisited in view of a gap in the classic work of H. Kunita (1971). We give new bounds for the exponential stability rates, which do not depend on the observations. In the ..."
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Cited by 33 (15 self)
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Abstract. Stability problem of the Wonham filter with respect to initial conditions is addressed. The case of ergodic signals is revisited in view of a gap in the classic work of H. Kunita (1971). We give new bounds for the exponential stability rates, which do not depend on the observations. In the nonergodic case, the stability is implied by identifiability conditions, formulated explicitly in terms of the transition intensities matrix and the observation structure. Key words. Nonlinear filtering, stability, Wonham filter
On the Optimality of Symbol by Symbol Filtering and Denoising
, 2003
"... We consider the problem of optimally recovering a finitealphabet discretetime stochastic process {X t } from its noisecorrupted observation process {Z t }. In general, the optimal estimate of X t will depend on all the components of {Z t } on which it can be based. We characterize nontrivial s ..."
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Cited by 19 (3 self)
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We consider the problem of optimally recovering a finitealphabet discretetime stochastic process {X t } from its noisecorrupted observation process {Z t }. In general, the optimal estimate of X t will depend on all the components of {Z t } on which it can be based. We characterize nontrivial situations (i.e., beyond the case where (X t , Z t ) are independent) for which optimum performance is attained using "symbol by symbol" operations (a.k.a.
Asymptotic stability, ergodicity and other asymptotic properties of the nonlinear
, 2003
"... ABSTRACT. – In this work we study connections between various asymptotic properties of the nonlinear filter. It is assumed that the signal has a unique invariant probability measure. The key property of interest is expressed in terms of a relationship between the observation σ field and the tail σ f ..."
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Cited by 10 (0 self)
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ABSTRACT. – In this work we study connections between various asymptotic properties of the nonlinear filter. It is assumed that the signal has a unique invariant probability measure. The key property of interest is expressed in terms of a relationship between the observation σ field and the tail σ field of the signal, in the stationary filtering problem. This property can be viewed as the permissibility of the interchange of the order of the operations of maximum and countable intersection for certain σfields. Under suitable conditions, it is shown that the above property is equivalent to various desirable properties of the filter such as (a) uniqueness of invariant measure for the signal, (b) uniqueness of invariant measure for the pair (signal, filter), (c) a finite memory property of the filter, (d) a property of finite time dependence between the signal and observation σ fields and (e) asymptotic stability of the filter. Previous works on the asymptotic stability of the filter for a variety of filtering models then identify a rich class of filtering problems for which the above equivalent properties hold. 2003 Éditions scientifiques et médicales Elsevier SAS
Monte Carlo algorithms and asymptotic problems in nonlinear filtering
 To Appear in Stochastics in Finite/Infinite Dimensions (Volume in honor of Gopinath Kallianpur
, 1999
"... This paper is an extension of [4], which dealt with a wide variety of approximations to optimal nonlinear filters over long time intervals, where ..."
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Cited by 9 (3 self)
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This paper is an extension of [4], which dealt with a wide variety of approximations to optimal nonlinear filters over long time intervals, where
Ergodic properties of the Nonlinear Filter
 Stochastic Processes and their Applications, 95:1–24
, 2000
"... 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 FellerMarkov process with values in a complete separable metric space E then th ..."
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Cited by 2 (1 self)
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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 FellerMarkov 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...
A Partial History of the Early Development of ContinuousTime Nonlinear Stochastic Systems Theory
"... The late 1950’s throughout the mid 1970’s were a period of renaissance in control theory. The classical theory, largely based on linear systems, Fourier and Laplace transform methods, and stability based on Bode and Nyquist plots, and RouthHurwitz criteria, was very successful, and provided the fou ..."
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The late 1950’s throughout the mid 1970’s were a period of renaissance in control theory. The classical theory, largely based on linear systems, Fourier and Laplace transform methods, and stability based on Bode and Nyquist plots, and RouthHurwitz criteria, was very successful, and provided the foundations
STABILITY OF NONLINEAR FILTERS: A SURVEY
"... Abstract. Filtering deals with the optimal estimation of signals from their noisy observations. The standard setting consists of a pair of random processes (X, Y) = (Xt, Yt)t≥0, where the signal component X is to be estimated at a current time t> 0 on the basis of the trajectory of Y, observed u ..."
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Abstract. Filtering deals with the optimal estimation of signals from their noisy observations. The standard setting consists of a pair of random processes (X, Y) = (Xt, Yt)t≥0, where the signal component X is to be estimated at a current time t> 0 on the basis of the trajectory of Y, observed up to t. Under the minimal mean square error criterion, the optimal estimate of Xt is the conditional expectation E(XtY[0,t]). If both X and (X, Y) are Markov processes, then the conditional distribution πt(A) = P (Xt ∈ AY[0,t]), A ⊆ R satisfies a recursive equation, called filter, which realizes the optimal fusion of the a priori statistical knowledge about the signal and the a posteriori information borne by the observation path. The filtering equation is to be initialized by the probability distribution ν of the signal at time t = 0. Suppose ν is unknown and another reasonable probability distribution ¯ν is used to start the filter. As the corresponding solution ¯πt(·) differs from the optimal πt(·), the natural question of stability arises: what are the conditions in terms of the signal/observation parameters to guarantee limt→ ∞ ‖πt − ¯πt ‖ = 0 in an appropriate sense? The article discusses the recent progress in solving this stability problem, which turns