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Exponential Stability for Nonlinear Filtering
, 1996
"... We study the a.s. exponential stability of the optimal filter w.r.t. its initial conditions. A bound is provided on the exponential rate (equivalently, on the memory length of the filter) for a general setting both in discrete and in continuous time, in terms of Birkhoff's contraction coefficient. C ..."
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Cited by 53 (2 self)
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We study the a.s. exponential stability of the optimal filter w.r.t. its initial conditions. A bound is provided on the exponential rate (equivalently, on the memory length of the filter) for a general setting both in discrete and in continuous time, in terms of Birkhoff's contraction coefficient. Criteria for exponential stability and explicit bounds on the rate are given in the specific cases of a diffusion process on a compact manifold, and discrete time Markov chains on both continuous and discretecountable state spaces. R'esum'e Nous 'etudions la stabilit'e du filtre optimal par raport `a ses conditions initiales. Le taux de d'ecroissance exponentielle est calcul'e dans un cadre g'en'eral, pour temps discret et temps continu, en terme du coefficient de contraction de Birkhoff. Des crit`eres de stabilit'e exponentielle et des bornes explicites sur le taux sont calcul'ees pour les cas particuliers d'une diffusion sur une vari'ete compacte, ainsi que pour des chaines de Markov sur ...
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.
Convergence of Empirical Processes for Interacting Particle Systems with Applications to Nonlinear Filtering
 Journal of Theoret. Probability
, 2000
"... In this paper, we investigate the convergence of empirical processes for a class of interacting particle numerical schemes arising in biology, genetic algorithms and advanced signal processing. The GlivenkoCantelli and Donsker theorems presented in this work extend the corresponding statements in t ..."
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Cited by 6 (3 self)
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In this paper, we investigate the convergence of empirical processes for a class of interacting particle numerical schemes arising in biology, genetic algorithms and advanced signal processing. The GlivenkoCantelli and Donsker theorems presented in this work extend the corresponding statements in the classical theory and apply to a class of genetic type particle numerical schemes of the nonlinear filtering equation. Keywords : Empirical processes, Interacting particle systems, GlivenkoCantelli and Donsker theorems. code A.M.S : 60G35, 92D25 UMR C55830, CNRS, Univ. PaulSabatier, 31062 Toulouse, delmoral@cict.fr y ledoux@cict.fr 1 Introduction 1.1 Background and Motivations Let E be a Polish space endowed with its Borel oefield B(E). We denote by M 1 (E) the space of all probability measures on E equipped with the weak topology. We recall that the weak topology is generated by the bounded continuous functions on E and we denote by C b (E) the space of these functions. Let O...
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.
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 up t ..."
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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