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Parameter estimation and asymptotic stability in stochastic filtering

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  • Papavasiliou, Anastasia

Abstract

In this paper, we study the problem of estimating a Markov chain X (signal) from its noisy partial information Y, when the transition probability kernel depends on some unknown parameters. Our goal is to compute the conditional distribution process , referred to hereafter as the optimal filter. Following a standard Bayesian technique, we treat the parameters as a non-dynamic component of the Markov chain. As a result, the new Markov chain is not going to be mixing, even if the original one is. We show that, under certain conditions, the optimal filters are still going to be asymptotically stable with respect to the initial conditions. Thus, by computing the optimal filter of the new system, we can estimate the signal adaptively.

Suggested Citation

  • Papavasiliou, Anastasia, 2006. "Parameter estimation and asymptotic stability in stochastic filtering," Stochastic Processes and their Applications, Elsevier, vol. 116(7), pages 1048-1065, July.
    • Handle: RePEc:eee:spapps:v:116:y:2006:i:7:p:1048-1065
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. Christophe Andrieu & Arnaud Doucet, 2002. "Particle filtering for partially observed Gaussian state space models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 64(4), pages 827-836, October.
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    Cited by:

    1. 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.
    2. Douc, Randal & Olsson, Jimmy & Roueff, François, 2020. "Posterior consistency for partially observed Markov models," Stochastic Processes and their Applications, Elsevier, vol. 130(2), pages 733-759.
    3. Jian He & Asma Khedher & Peter Spreij, 2021. "A Kalman particle filter for online parameter estimation with applications to affine models," Statistical Inference for Stochastic Processes, Springer, vol. 24(2), pages 353-403, July.

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