Ergodicity of filtering processes: the history of a mistake and attempts to correct it

Łukasz Stettner

Published: 2013-09-30

Vol. 27 (2013)

Ergodicity of filtering processes: the history of a mistake and attempts to correct it

Łukasz Stettner

Abstract

The paper describes briefly a history of filtering problems of Markov processes and then concentrates on ergodic properties of filtering process. A mistake in a famous Kunita paper on ergodicity of filtering processes is shown. Then the paper reviews various attempts trying to correct this mistake.

Keywords:

Markov processes , partial observation , filtering process , invariant measures

Download files

PDF

Citation rules

Stettner, Łukasz. (2013). Ergodicity of filtering processes: the history of a mistake and attempts to correct it. Annales Mathematicae Silesianae, 27, 39–58. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/13999

Similar Articles

Jason Bentley, Piotr Mikusiński, Transfunctions applied to plans, Markov operators and optimal transport , Annales Mathematicae Silesianae: 2024: Online First
Dariusz Dudzik, Marcin Skrzyński, Outer measures on a commutative ring induced by measures on its spectrum , Annales Mathematicae Silesianae: Vol. 31 (2017)
Piotr Mikusiński, John Paul Ward, On the Radon-Nikodym property for vector measures and extensions of transfunctions , Annales Mathematicae Silesianae: Vol. 35 No. 1 (2021)
McSylvester Ejighikeme Omaba, Eze R. Nwaeze, Generalized fractional inequalities of the Hermite-Hadamard type for convex stochastic processes , Annales Mathematicae Silesianae: Vol. 35 No. 1 (2021)
Maria C. Mariani, Osei K. Tweneboah, Miguel A. Valles, Pavel Bezdek, Complex Gleason measures and the Nemytsky operator , Annales Mathematicae Silesianae: Vol. 33 (2019)
Piotr Gwiżdż, Applications of stochastic semigroups to queueing models , Annales Mathematicae Silesianae: Vol. 33 (2019)
Maciej Ślęczka, Exponential convergence for Markov systems , Annales Mathematicae Silesianae: Vol. 29 (2015)
Joanna Kubieniec, The law of the iterated logarithm for random dynamical system with jumps and state-dependent jump intensity , Annales Mathematicae Silesianae: Vol. 35 No. 2 (2021)
Kazimierz Nikodem, On some properties of quadratic stochastic processes , Annales Mathematicae Silesianae: Vol. 3 (1990)
Aleksander Weron, Mathematical models for dynamics of molecular processes in living biological cells. A single particle tracking approach , Annales Mathematicae Silesianae: Vol. 32 (2018)

1 2 3 4 5 > >>

You may also start an advanced similarity search for this article.

References

1. Atar R., Zeitouni O., Lyapunov exponents for finite-state nonlinear filtering, SIAM J. Control Optim. 35 (1997), 36–55.
2. Baxendale P., Chigansky P., Liptser R., Asymptotic stability of the Wonham filter: ergodic and nonergodic signals, Preprint 2002.
3. Borkar V.S., Ergodic control of partially observed Markov chains, Systems Control Lett. 34 (1998), 185–189.
4. Borkar V.S., Budhiraja A., A further remark on dynamic programming for partially observed Markov processes, Stochastic Process. Appl. 112 (2004), 79–93.
5. Di Masi G., Stettner L., Ergodicity of Hidden Markov Models, Math. Control Signals Systems 17 (2005), 269–296.
6. Di Masi G., Stettner L., Ergodicity of filtering process by vanishing discount approach, Systems Control Lett. 57 (2008), 150–157.
7. Di Masi G., Stettner L., Risk sensitive control of discrete time partially observed Markov processes with infinite horizon, Stochastics and Stochastics Rep. 67 (1999), 309–322.
8. Ikeda N., Watanabe Sh., Stochastic Differential Equations and Diffusion Processes, North-Holland, Amsterdam, 1981.
9. Kaijser T., A limit theorem for partially observed Markov chains, Ann. Probab. 3 (1975), 677–696.
10. Kakutani S., Ergodic theorems and the Markoff processes with a stable distribution, Proc. Imp. Acad. Tokyo 16 (1940), 49–54.
11. Kallenberg O., Foundations of Modern Probability, Springer-Verlag, New York–Berlin, 1997.
12. Kunita H., Asymptotic behaviour of the nonlinear filtering errors of Markov process, J. Multivariate Anal. 1 (1971), 365–393.
13. Kunita H., Ergodic properties of nonlinear filtering processes , in: Spatial Stochastic Processes, ed. by Aleksander K.C., Watkins J.C., Progr. Probab. 19 , Birkhauser, Boston, 1991, pp. 233–256.
14. Liptser R., Shiryaev A.N., Statistics of Random Processes II. Applications, Second Edition, Springer-Verlag, New York–Berlin, 2001.
15. Liverani C., Decay of correlations, Ann. of Math. 142 (1995), 239–301.
16. Meyn S.P., Tweedie R.L., Markov Chains and Stochastics Stability, Springer-Verlag, New York–Berlin, 1993.
17. Ocone D., Pardoux E., Asymptotic stability of the optimal filter with respect to its initial conditions, SIAM J. Control Optim. 34 (1996), 226–243.
18. Runggaldier W., Stettner L., Approximations of Discrete Time Partially Observed Control Problems, Applied Mathematics Monographs CNR, Giardini Editori, Pisa, 1994.
19. Schäl M., Average Optimality in Dymanic Programming with General State Space, Math. Oper. Res. 18 (1993), 163–172.
20. Stettner L., On Invariant Measures of Filtering Processes , in: Proc. 4th Bad Honnef Conf. on Stochastic Differential Systems, ed. by Christopeit N., Helmes K., Kohlmann M., Lect. Notes in Control Inf. Sci. 126, Springer-Verlag, New York–Berlin, 1989, pp. 279–292.
21. Stettner L., Ergodic control of partially observed Markov processes with equivalent transition probabilities, Appl. Math. (Warsaw) 22 (1993), 25–38.
22. Tong X.T., van Handel R., Ergodicity and stability of the conditional distributions of nondegenerate Markov chains, Ann. Appl. Probab. 22 (2012), 1495–1540.
23. van Handel R., The stability of conditional Markov processes and Markov chains in random environments, Ann. Probab. 37 (2009), 1876–1925.
24. van Handel R., A nasty filtering problem, Preprint 2010, arXiv:1009.0507.
25. Yosida K., Functional Analysis, Springer-Verlag, New York–Berlin, 1978. Google Scholar

Vol. 27 (2013)
Published: 2013-09-30

ISSN: 0860-2107

eISSN: 2391-4238

10.1515/amsil

Publisher

University of Silesia Press

Submit