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18
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 25 (13 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
Robustness of Nonlinear Filters over the Infinite Time Interval
 SIAM J. on Control and Optimization
, 1997
"... Nonlinear filtering is one of the classical areas of stochastic control. From the point of view of practical usefulness, it is important that the filter not be too sensitive to the assumptions made on the initial distribution, the transition function of the underlying signal process and the model fo ..."
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Cited by 18 (7 self)
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Nonlinear filtering is one of the classical areas of stochastic control. From the point of view of practical usefulness, it is important that the filter not be too sensitive to the assumptions made on the initial distribution, the transition function of the underlying signal process and the model for the observation. This is particularly acute if the filter is of interest over a very long or potentially infinite time interval. Then the effects of small errors in the model which is used to construct the filter might accumulate to make the output useless for large time. The problem of asymptotic sensitivity to the initial condition has been treated in several papers. We are concerned with this as well as with the sensitivity to the signal model, uniformly over the infinite time interval. It is conceivable that the effects of even small errors in the model will accumulate so that the filter will eventually be useless. The robustness is shown for three classes of problems. For the first tw...
Markov Property and Ergodicity of the Nonlinear Filter
, 1999
"... In this paper we first prove, under quite general conditions, that the nonlinear filter and the pair: (signal,filter) are FellerMarkov processes. The state space of the signal is allowed to be non locally compact and the observation function: h can be unbounded. Our proofs in contrast to those of K ..."
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Cited by 11 (4 self)
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In this paper we first prove, under quite general conditions, that the nonlinear filter and the pair: (signal,filter) are FellerMarkov processes. The state space of the signal is allowed to be non locally compact and the observation function: h can be unbounded. Our proofs in contrast to those of Kunita(1971,1991), Stettner(1989) do not depend upon the uniqueness of the solutions to the filtering equations. We then obtain conditions for existence and uniqueness of invariant measures for the nonlinear filter and the pair process. These results extend those of Kunita and Stettner, which hold for locally compact state space and bounded h, to our general framework. Finally we show that the recent results of OconePardoux [11] on asymptotic stability of the nonlinear filter, which use the KunitaStettner setup, hold for the general situation considered in this paper. Key Words: nonlinear filtering, invariant measures, asymptotic stability, measure valued processes. AMS Classification:60 ...
Robustness of Zakai's equation via FeynmanKac representations
 Stochastic Analysis, Control, Optimization and Applications : A Volume in Honor of W.H. Fleming, Systems & Control : Foundations & Applications
, 1997
"... We propose to study the sensitivity of the optimal filter to its initialization, by looking at the distance between two differently initialized filtering processes in terms of the ratio between two simple FeynmanKac integrals in the product space. We illustrate, by considering two simple examples, ..."
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Cited by 7 (1 self)
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We propose to study the sensitivity of the optimal filter to its initialization, by looking at the distance between two differently initialized filtering processes in terms of the ratio between two simple FeynmanKac integrals in the product space. We illustrate, by considering two simple examples, how this approach may be employed to study the asymptotic decay rate, as the difference between the growth rates of the two integrals. We apply asymptotic methods, such as large deviations, to estimate these growth rates. The examples we consider are the linear case, where we recover known results, and a case where the drift term in the state process is nonlinear. In both cases, only the small noise regime and only onedimensional diffusions are studied.
Model robustness of finite state nonlinear filtering over the infinite time horizon
 Ann. Appl. Probab
"... Abstract. We investigate the robustness of nonlinear filtering for continuous time finite state Markov chains, observed in white noise, with respect to misspecification of the model parameters. It is shown that the distance between the optimal filter and that with incorrect model parameters converge ..."
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Cited by 5 (5 self)
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Abstract. We investigate the robustness of nonlinear filtering for continuous time finite state Markov chains, observed in white noise, with respect to misspecification of the model parameters. It is shown that the distance between the optimal filter and that with incorrect model parameters converges to zero uniformly over the infinite time interval as the misspecified model converges to the true model, provided the signal obeys a mixing condition. The filtering error is controlled through the exponential decay of the derivative of the nonlinear filter with respect to its initial condition. We allow simultaneously for misspecification of the initial condition, of the transition rates of the signal, and of the observation function. The first two cases are treated by relatively elementary means, while the latter case requires the use of Skorokhod integrals and tools of anticipative stochastic calculus. 1.
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 5 (2 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
Uniform time average consistency of Monte Carlo particle filters
 BP 101  54602 VillerslèsNancy Cedex Centre de recherche INRIA Paris – Rocquencourt : Domaine de Voluceau  Rocquencourt  BP 105  78153 Le Chesnay Cedex Centre de recherche INRIA Rennes – Bretagne Atlantique : IRISA, Campus universitaire de Beaulieu 
, 2009
"... Abstract. We prove that bootstrap type Monte Carlo particle filters approximate the optimal nonlinear filter in a time average sense uniformly with respect to the time horizon when the signal is ergodic and the particle system satisfies a tightness property. The latter is satisfied without further a ..."
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Cited by 5 (0 self)
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Abstract. We prove that bootstrap type Monte Carlo particle filters approximate the optimal nonlinear filter in a time average sense uniformly with respect to the time horizon when the signal is ergodic and the particle system satisfies a tightness property. The latter is satisfied without further assumptions when the signal state space is compact, as well as in the noncompact setting when the signal is geometrically ergodic and the observations satisfy additional regularity assumptions. 1.
Asymptotic Stability, Ergodicity and Other Asymptotic Properties of the Nonlinear Filter
, 2002
"... 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 # field and the tail # field of the ..."
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Cited by 4 (0 self)
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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 # field and the tail # 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 # fields. 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 # 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.
Robustness and Convergence of Approximations to Nonlinear Filters for JumpDiffusions
 Computational and Applied Math
, 1996
"... The paper treats numerical approximations to the nonlinear filtering problem for jumpdiffusion processes. This is a key problem in stochastic systems analysis. The processes are defined, and the optimal filters described. In the general nonlinear case, the optimal filters cannot be computed, and s ..."
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Cited by 4 (2 self)
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The paper treats numerical approximations to the nonlinear filtering problem for jumpdiffusion processes. This is a key problem in stochastic systems analysis. The processes are defined, and the optimal filters described. In the general nonlinear case, the optimal filters cannot be computed, and some numerical approximation is needed. Then the weak conditions that are required for the convergence of the approximations are given and the convergence is proved. Examples of useful approximations which satisfy the conditions are given. Quite weak conditions are given under which the approximating filter is continuous in the observation function, and it is shown that our canonical methods satisfy the conditions. Such continuity is essential if the approximations are to be used with confidence on actual physical data. Finally, we prove the convergence of monte carlo methods for approximating the optimal filters, and also show that the optimal filter is continuous in the parameters of the si...
A Uniformly Convergent Adaptive Particle Filter. Preprint: http://www.princeton.edu/~ap/uniform.pdf
"... Particle filters are MonteCarlo methods that aim to approximate the optimal filter of a partially observed Markov chain. In this paper, we study the case where the transition kernel of the Markov chain depends on unknown parameters: we construct a particle filter for the simultaneous estimation of ..."
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Cited by 4 (1 self)
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Particle filters are MonteCarlo methods that aim to approximate the optimal filter of a partially observed Markov chain. In this paper, we study the case where the transition kernel of the Markov chain depends on unknown parameters: we construct a particle filter for the simultaneous estimation of the parameter and the partially observed Markov chain (adaptive estimation) and we prove the convergence of this filter to the correct optimal filter, as time and the number of particles go to infinity. The filter presented here generalizes Del Moral’s MonteCarlo particle filter, presented in [5].