Asymptotic Stability of the Wonham Filter: Ergodic and Nonergodic Signals

The 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.

[1]  W. Wonham.  Some applications of stochastic difierential equations to optimal nonlinear ltering , 1964 .

[2]  H. Kunita.  Asymptotic behavior of the nonlinear filtering errors of Markov processes , 1971 .

[3]  D. Vere-Jones.  Markov Chains , 1972, Nature.

[4]  Thomas Kaijser.  A Limit Theorem for Partially Observed Markov Chains , 1975 .

[5]  V. Benes,et al.  Estimation and control for linear, partially observable systems with non-gaussian initial distribution☆☆☆ , 1983 .

[6]  V. Benes,et al.  Estimation and control for linear, partially observable systems with non-gaussian initial distribution☆☆☆ , 1983 .

[7]  H. Weizsäcker.  Exchanging the order of taking suprema and countable intersections of $\sigma $-algebras , 1983 .

[8]  A. Makowski.  Filtering formulae for partially observed linear systems with non-gaussian initial conditions , 1986 .

[9]  Decision Systems.,et al.  Lyapunov Exponents for Filtering Problems , 1988 .

[10]  Y. Sinai.  Kolmogorov's Work on Ergodic Theory , 1989 .

[11]  Alʹbert Nikolaevich Shiri︠a︡ev,et al.  Theory of martingales , 1989 .

[12]  Ł. Stettner.  Invariant measures of the pair: state, approximate filtering process , 1991 .

[13]  David Williams,et al.  Probability with Martingales , 1991, Cambridge mathematical textbooks.

[14]  H. Kunita.  Ergodic Properties of Nonlinear Filtering Processes , 1991 .

[15]  A. Makowski,et al.  Discrete-time filtering for linear systems with non-Gaussian initial conditions: asymptotic behavior of the difference between the MMSE and LMSE estimates , 1992 .

[16]  D. Ocone,et al.  Asymptotic Stability of the Optimal Filter with Respect toIts Initial Condition , 1996 .

[17]  A. Budhiraja,et al.  Exponential stability of discrete-time filters for bounded observation noise , 1997 .

[18]  R. Atar,et al.  Exponential stability for nonlinear filtering , 1997 .

[19]  C. SIAMJ..  LYAPUNOV EXPONENTS FOR FINITE STATE NONLINEAR FILTERING , 1997 .

[20]  J. Lynch,et al.  A weak convergence approach to the theory of large deviations , 1997 .

[21]  R. Atar.  Exponential stability for nonlinear filtering of diffusion processes in a noncompact domain , 1998 .

[22]  H. Kushner,et al.  Robustness of Nonlinear Filters Over the Infinite Time Interval , 1998 .

[23]  G. Prato,et al.  Asymptotic Ergodicity of the Process of Conditional Law in Some Problem of Non-linear Filtering , 1999 .

[24]  H. Kushner,et al.  Approximation and Limit Results for Nonlinear Filters Over an Infinite Time Interval , 1999 .

[25]  R. Atar,et al.  Robustness of Zakai’s Equation via Feynman-Kac Representations , 1999 .

[26]  D. Ocone.  Asymptotic stability of beneš filters , 1999 .

[27]  A. Budhiraja,et al.  Exponential stability in discrete-time filtering for non-ergodic signals , 1999 .

[28]  P. Moral,et al.  On the stability of measure valued processes with applications to filtering , 1999 .

[29]  D. Ocone.  Entropy Inequalities and Entropy Dynamics in Nonlinear Filtering of Diffusion Processes , 1999 .

[30]  J. M. C. Clark,et al.  Relative Entropy and Error Bounds for Filtering of Markov Processes , 1999, Math. Control. Signals Syst..

[31]  Alain Le Breton,et al.  Asymptotic Optimality of Approximate Filters in Stochastic Systems with Colored Noises , 2000, SIAM J. Control. Optim..

[32]  Laurent Mevel,et al.  Exponential Forgetting and Geometric Ergodicity in Hidden Markov Models , 2000, Math. Control. Signals Syst..

[33]  Harold J. Kushner,et al.  Approximation and Limit Results for Nonlinear Filters Over an Infinite Time Interval: Part II, Random Sampling Algorithms , 2000, SIAM J. Control. Optim..

[34]  Frédéric Cérou,et al.  Long Time Behavior for Some Dynamical Noise Free Nonlinear Filtering Problems , 2000, SIAM J. Control. Optim..

[35]  Amarjit Budhiraja,et al.  Markov Property and Ergodicity of the Nonlinear Filter , 2000, SIAM J. Control. Optim..

[36]  F. LeGland,et al.  Stability and approximation of nonlinear filters in the Hilbert metric, and application to particle filters , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).

[37]  P. Moral,et al.  On the stability of interacting processes with applications to filtering and genetic algorithms , 2001 .

[38]  A. Budhiraja.  Ergodic properties of the nonlinear filter , 2001 .

[39]  P. Moral,et al.  On the stability of nonlinear Feynman-Kac semigroups , 2002 .

[40]  A. Budhiraja.  On invariant measures of discrete time filters in the correlated signal-noise case , 2002 .

[41]  F. Gland,et al.  STABILITY AND UNIFORM APPROXIMATION OF NONLINEAR FILTERS USING THE HILBERT METRIC AND APPLICATION TO PARTICLE FILTERS1 , 2004 .

[42]  F. LeGland,et al.  A robustification approach to stability and to uniform particle approximation of nonlinear filters: the example of pseudo-mixing signals , 2003 .

[43]  A. Shiryaev,et al.  Probability (2nd ed.) , 1995, Technometrics.