Exponential stability for nonlinear filtering

Rami Atar; Ofer Zeitouni

Annales de l'I.H.P. Probabilités et statistiques (1997)

  • Volume: 33, Issue: 6, page 697-725
  • ISSN: 0246-0203

How to cite


Atar, Rami, and Zeitouni, Ofer. "Exponential stability for nonlinear filtering." Annales de l'I.H.P. Probabilités et statistiques 33.6 (1997): 697-725. <http://eudml.org/doc/77587>.

author = {Atar, Rami, Zeitouni, Ofer},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {nonlinear filtering; nonlinear smoothing; exponential stability; Birkhoff contraction coefficient},
language = {eng},
number = {6},
pages = {697-725},
publisher = {Gauthier-Villars},
title = {Exponential stability for nonlinear filtering},
url = {http://eudml.org/doc/77587},
volume = {33},
year = {1997},

AU - Atar, Rami
AU - Zeitouni, Ofer
TI - Exponential stability for nonlinear filtering
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 1997
PB - Gauthier-Villars
VL - 33
IS - 6
SP - 697
EP - 725
LA - eng
KW - nonlinear filtering; nonlinear smoothing; exponential stability; Birkhoff contraction coefficient
UR - http://eudml.org/doc/77587
ER -


  1. [1] R. Atar and O. Zeitouni, Lyapunov Exponents for Finite State Nonlinear Filtering, Siam J. Contr. Optim., Vol. 35, No 1, 1997, pp. 36-55. Zbl0940.93073MR1430282
  2. [2] G. Birkhoff, Extensions of Jentzsch's Theorem, Trans. Am. Math. Soc., Vol. 85, 1957, pp. 219-227. Zbl0079.13502MR87058
  3. [3] G. Birkhoff, Lattice Theory, Am. Math. Soc. Publ.25, 3rd ed., 1967. Zbl0153.02501
  4. [4] P. Bougerol, Théorèmes limites pour les systèmes linéaires à coefficients markoviens, Prob. Theory Rel. Fields, Vol. 78, 1988, pp. 192-221. Zbl0627.60054MR945109
  5. [5] P. Bougerol and J. Lacroix, Products of Random Matrices with Applications to Schrödinger Operators, Birkhäuser, 1985. Zbl0572.60001MR886674
  6. [6] E.B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, 1989. Zbl0699.35006MR990239
  7. [7] B. Delyon and O. Zeitouni, Lyapunov exponents for filtering problems, Applied Stochastic Analysis, edited by Davis, M. H. A. and Elliot, R. J., Gordon and Breach Science Publishers, London, pp. 511-521, 1991. Zbl0738.60033MR1108433
  8. [8] F. Flandoli and K. Schaumlöffel, Stochastic Parabolic Equations in Bounded Domains: Random Evolution Operator and Lyapunov Exponents, Stochastics and Stochastic Reports, Vol. 29, 1990, pp. 461-485. Zbl0704.60060MR1124162
  9. [9] E. Hopf, An Inequality for Positive Linear Integral Operators, Journal of Mathematics and Mechanics, Vol. 12, No. 5, 1963, pp. 683-692. Zbl0115.32501MR165325
  10. [10] I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus, Springer-Verlag, 1991. Zbl0734.60060MR1121940
  11. [11] S. Karlin, Total Positivity, Stanford University Press, Stanford, California, 1968. Zbl0219.47030MR230102
  12. [12] H. Kunita, Asymptotic Behavior of the Nonlinear Filtering Errors of Markov Processes, J. Multivariate Anal., Vol. 1, 1971, pp. 365-393. Zbl0245.93027MR301812
  13. [13] R.S. Liptser and A.N. Shiryayev, A. N. Statistics of Random Processes, Nauka, Moscow, 1974, English ed., Springer-Verlag, New-york, 1977. Zbl0364.60004
  14. [14] D. Ocone and E. Pardoux, Asymptotic Stability of the Optimal Filter with respect to its Initial Condition, Siam J. Contr. Optim., Vol. 34, No. 1, 1996, pp. 226-243. Zbl1035.93508MR1372912
  15. [15] K.R. Parthasarathy, Probability Measures on Metric Spaces, Academic Press, New York and London, 1967. Zbl0153.19101MR226684
  16. [16] Y. Peres, Domains of Analytic Continuation for the Top Lyapunov Exponent, Ann. Inst. Henri Poincaré, Vol. 28, 1992, No. 1, pp. 131-148. Zbl0794.58023MR1158741
  17. [17] E. Seneta, Non-negative Matrices and Markov Chains, Springer-Verlag, 1981. Zbl0471.60001
  18. [18] L. Stettner, On Invariant Measures of Filtering Processes, Stochastic Differential Systems, Proc. 4th Bad Honnef Conf., 1988, Lecture Notes in Control and Inform. Sci.126, edited by Christopeit, N., Helmes, K. and Kohlmann, M., Springer, 1989, pp. 279-292. Zbl0683.93082MR1236074
  19. [19] L. Stettner, Invariant Measures of the Pair: State, Approximate Filtering Process, Colloq. Math., LXII, 1991, pp. 347-351. Zbl0795.60028MR1142935
  20. [20] M. Zakai, On the Optimal Filtering of Diffusion Processes, Z. Wahr. Verw. Geb., Vol. 11, 1969, pp. 230-243. Zbl0164.19201MR242552
  21. [21] O. Zeitouni and B.Z. Bobrovsky, On the Joint Nonlinear Filtering-Smoothing of Diffusion Processes, Systems & Control Letters, Vol. 7, 1986, pp. 317-321. Zbl0661.93069

Citations in EuDML Documents

  1. Benjamin Favetto, On the asymptotic variance in the central limit theorem for particle filters
  2. Pierre Del Moral, Laurent Miclo, Branching and interacting particle systems. Approximations of Feynman-Kac formulae with applications to non-linear filtering
  3. Pierre Del Moral, Alice Guionnet, On the stability of interacting processes with applications to filtering and genetic algorithms
  4. Benjamin Favetto, On the asymptotic variance in the central limit theorem for particle filters
  5. A. Budhiraja, Asymptotic stability, ergodicity and other asymptotic properties of the nonlinear filter

NotesEmbed ?


You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.


Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.