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Asymptotic behavior of the nonlinear filtering errors of Markov processes

Author

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  • Kunita, Hiroshi

Abstract

Nonlinear filtering process [pi]t, t >= 0 of a Markovian signal process with the state space S is regarded as a stochastic process with values in the set of all probability distributions over S. Under a suitable condition, it is shown that the filtering process is Markovian and that the invariant measure of the filtering process exists uniquely if and only if the stationary signal process (flow) is purely nondeterministic. These results are applied to the study for the asymptotic behavior of the filtering error. It turns out that the minimal asymptotic error is 0 if the signal process is transient, null recurrent or deterministic positive recurrent.

Suggested Citation

  • Kunita, Hiroshi, 1971. "Asymptotic behavior of the nonlinear filtering errors of Markov processes," Journal of Multivariate Analysis, Elsevier, vol. 1(4), pages 365-393, December.
    • Handle: RePEc:eee:jmvana:v:1:y:1971:i:4:p:365-393
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    Citations

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    Cited by:

    1. Deck, T., 2006. "Asymptotic properties of Bayes estimators for Gaussian Itô-processes with noisy observations," Journal of Multivariate Analysis, Elsevier, vol. 97(2), pages 563-573, February.
    2. Ying Jiao, 2009. "Multiple defaults and contagion risks," Papers 0912.3132, arXiv.org.
    3. Budhiraja, A., 2001. "Ergodic properties of the nonlinear filter," Stochastic Processes and their Applications, Elsevier, vol. 95(1), pages 1-24, September.
    4. Moral, P. Del & Guionnet, A., 1998. "Large deviations for interacting particle systems: Applications to non-linear filtering," Stochastic Processes and their Applications, Elsevier, vol. 78(1), pages 69-95, October.
    5. Papavasiliou, Anastasia, 2006. "Parameter estimation and asymptotic stability in stochastic filtering," Stochastic Processes and their Applications, Elsevier, vol. 116(7), pages 1048-1065, July.
    6. Budhiraja, A. & Ocone, D., 1999. "Exponential stability in discrete-time filtering for non-ergodic signals," Stochastic Processes and their Applications, Elsevier, vol. 82(2), pages 245-257, August.
    7. Ying Jiao, 2009. "Multiple defaults and contagion risks," Working Papers hal-00441500, HAL.
    8. P. Del Moral & M. Ledoux, 2000. "Convergence of Empirical Processes for Interacting Particle Systems with Applications to Nonlinear Filtering," Journal of Theoretical Probability, Springer, vol. 13(1), pages 225-257, January.
    9. LeGland, François & Oudjane, Nadia, 2003. "A robustification approach to stability and to uniform particle approximation of nonlinear filters: the example of pseudo-mixing signals," Stochastic Processes and their Applications, Elsevier, vol. 106(2), pages 279-316, August.
    10. Beatris Adriana Escobedo-Trujillo & Javier Garrido-Meléndez & Gerardo Alcalá & J. D. Revuelta-Acosta, 2022. "Optimal Control with Partially Observed Regime Switching: Discounted and Average Payoffs," Mathematics, MDPI, vol. 10(12), pages 1-28, June.

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