IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v95y2001i1p1-24.html
   My bibliography  Save this article

Ergodic properties of the nonlinear filter

Author

Listed:
  • Budhiraja, A.

Abstract

In a recent work (Bhatt et al., SIAM J. Control Optim. 39 (2000) 928) various Markov and ergodicity properties of the nonlinear filter, for the classical model of nonlinear filtering, were studied. It was shown that under quite general conditions, when the signal is a Feller-Markov process with values in a complete separable metric space E then the pair process (signal, filter) is also a Feller-Markov process with state space , where is the space of probability measures on E. Furthermore, it was shown that if the signal has a unique invariant measure then, under appropriate conditions, uniqueness of the invariant measure for the above pair process holds within a certain restricted class of invariant measures. In many asymptotic problems concerning approximate filters (Budhiraja and Kushner, SIAM J. Control Optim. 37 (1997) 1946; 38 (2000) 1874) it is desirable to have the uniqueness of the invariant measure to hold in the class of all invariant measures. In this paper we first show that for a rich class of filtering problems, when the signal has a unique invariant measure, the property of "asymptotic stability" for the filter holds. Using this property of asymptotic stability we then provide sufficient conditions under which the (signal, filter) pair has a unique invariant measure. We also show that, in a certain sense, the property of asymptotic stability is necessary for the uniqueness of the invariant measure.

Suggested Citation

  • Budhiraja, A., 2001. "Ergodic properties of the nonlinear filter," Stochastic Processes and their Applications, Elsevier, vol. 95(1), pages 1-24, September.
  • Handle: RePEc:eee:spapps:v:95:y:2001:i:1:p:1-24
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304-4149(01)00090-4
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Karandikar, Rajeeva L., 1995. "On pathwise stochastic integration," Stochastic Processes and their Applications, Elsevier, vol. 57(1), pages 11-18, May.
    2. 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.
    3. 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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. Papavasiliou, Anastasia, 2006. "Parameter estimation and asymptotic stability in stochastic filtering," Stochastic Processes and their Applications, Elsevier, vol. 116(7), pages 1048-1065, July.
    3. Zhiqiang Li & Jie Xiong, 2015. "Stability of the filter with Poisson observations," Statistical Inference for Stochastic Processes, Springer, vol. 18(3), pages 293-313, October.
    4. Ying Jiao, 2009. "Multiple defaults and contagion risks," Working Papers hal-00441500, HAL.
    5. 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.
    6. Nutz, Marcel, 2015. "Robust superhedging with jumps and diffusion," Stochastic Processes and their Applications, Elsevier, vol. 125(12), pages 4543-4555.
    7. Dirk Becherer & Klebert Kentia, 2017. "Good Deal Hedging and Valuation under Combined Uncertainty about Drift and Volatility," Papers 1704.02505, arXiv.org.
    8. Jakv{s}a Cvitani'c & Dylan Possamai & Nizar Touzi, 2015. "Dynamic programming approach to principal-agent problems," Papers 1510.07111, arXiv.org, revised Jan 2017.
    9. Possamaï, Dylan, 2013. "Second order backward stochastic differential equations under a monotonicity condition," Stochastic Processes and their Applications, Elsevier, vol. 123(5), pages 1521-1545.
    10. Lin, Qian, 2019. "Jensen inequality for superlinear expectations," Statistics & Probability Letters, Elsevier, vol. 151(C), pages 79-83.
    11. Ying Jiao, 2009. "Multiple defaults and contagion risks," Papers 0912.3132, arXiv.org.
    12. 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.
    13. Felix-Benedikt Liebrich & Marco Maggis & Gregor Svindland, 2020. "Model Uncertainty: A Reverse Approach," Papers 2004.06636, arXiv.org, revised Mar 2022.
    14. Beißner, Patrick, 2013. "Coherent Price Systems and Uncertainty-Neutral Valuation," VfS Annual Conference 2013 (Duesseldorf): Competition Policy and Regulation in a Global Economic Order 80010, Verein für Socialpolitik / German Economic Association.
    15. Larry G. Epstein & Shaolin Ji, 2013. "Ambiguous Volatility and Asset Pricing in Continuous Time," The Review of Financial Studies, Society for Financial Studies, vol. 26(7), pages 1740-1786.
    16. Buckdahn, Rainer & Ma, Jin & Zhang, Jianfeng, 2015. "Pathwise Taylor expansions for random fields on multiple dimensional paths," Stochastic Processes and their Applications, Elsevier, vol. 125(7), pages 2820-2855.
    17. repec:hal:spmain:info:hdl:2441/5rkqqmvrn4tl22s9mc0ck8ecp is not listed on IDEAS
    18. Epstein, Larry G. & Ji, Shaolin, 2014. "Ambiguous volatility, possibility and utility in continuous time," Journal of Mathematical Economics, Elsevier, vol. 50(C), pages 269-282.
    19. Nicolas Perkowski & David J. Promel, 2013. "Pathwise stochastic integrals for model free finance," Papers 1311.6187, arXiv.org, revised Jun 2016.
    20. B. Acciaio & M. Beiglbock & F. Penkner & W. Schachermayer & J. Temme, 2012. "A trajectorial interpretation of Doob's martingale inequalities," Papers 1202.0447, arXiv.org, revised Jul 2013.
    21. repec:spo:wpmain:info:hdl:2441/5rkqqmvrn4tl22s9mc0ck8ecp is not listed on IDEAS
    22. repec:hal:wpspec:info:hdl:2441/5rkqqmvrn4tl22s9mc0ck8ecp is not listed on IDEAS
    23. Amine Ismail & Huy^en Pham, 2016. "Robust Markowitz mean-variance portfolio selection under ambiguous covariance matrix ," Papers 1610.06805, arXiv.org, revised Mar 2017.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:spapps:v:95:y:2001:i:1:p:1-24. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.