Exponential stability for nonlinear filtering

Annales de l'institut Henri Poincare (B) Probability and Statistics

Volumn 33, Issue 6, 1997, Pages 697-725

  Atar, Rami ^a   Zeitouni, Ofer ^a  

^a TECHNION ISRAEL INSTITUTE OF TECHNOLOGY   (Israel)

Author keywords

Birkhoff contraction coefficient; Exponential stability; Nonlinear filtering; Nonlinear smoothing

Indexed keywords

EID: 0031498168     PISSN: 02460203     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0246-0203(97)80110-0     Document Type: Article

References (21)

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.
2. [2] G. BIRKHOFF, Extensions of Jentzsch's Theorem, Trans. Am. Math. Soc., Vol. 85, 1957, pp. 219-227.
3. [3] G. BIRKHOFF, Lattice Theory, Am. Math. Soc. Publ. 25, 3rd ed., 1967.
4. [4] P. BOUGEROL, Théorèmes limites pour les systèmes linéaires à coefficients markoviens, Prob. Theory Rel. Fields, Vol. 78, 1988, pp. 192-221.
5. [5] P. BOUGEROL and J. LACROIX, Products of Random Matrices with Applications to Schrödinger Operators, Birkhäuser, 1985.
6. [6] E. B. DAVIES, Heat Kernels and Spectral Theory, Cambridge University Press, 1989.
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.
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.
9. [9] E. HOPF, An Inequality for Positive Linear Integral Operators, Journal of Mathematics and Mechanics, Vol. 12, No. 5, 1963, pp. 683-692.
10. [10] I. KARATZAS and S. E, SHREVE, Brownian Motion and Stochastic Calculus, Springer-Verlag, 1991.
11. [11] S. KARLIN, Total Positivity, Stanford University Press, Stanford, California, 1968.
12. [12] H. KUNITA, Asymptotic Behavior of the Nonlinear Filtering Errors of Markov Processes, J. Multivariate Anal., Vol. 1, 1971, pp. 365-393.
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.
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.
15. [15] K. R. PARTHASARATHY, Probability Measures on Metric Spaces, Academic Press, New York and London, 1967.
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.
17. [17] E. SENETA, Non-negative Matrices and Markov Chains, Springer-Verlag, 1981.
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.
19. [19] L. STETTNER, Invariant Measures of the Pair: State, Approximate Filtering Process, Colloq. Math., LXII, 1991, pp. 347-351.
20. [20] M. ZAKAI, On the Optimal Filtering of Diffusion Processes, Z. Wahr. Verw, Geb., Vol. 11, 1969, pp. 230-243.
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.