Ergodic Properties of Nonlinear Filtering Processes

  • Hiroshi Kunita
Part of the Progress in Probability book series (PRPR, volume 19)

Abstract

Let x t be a temporally homogenous Markov process with state space S, called a system process in this paper. Suppose that we want to observe the sample path x t , but what we can observe is a stochastic process Y t of the form
$$ Y_t = \int_0^t {h\left( {x_s } \right)dt + N_t ,} $$
(0.1)
where h is a continous function on S and N t is a standard Brownian motion independent of x s . The filtering of the system based on the observation data Y t is defined by a conditional distribution
$$ \pi _t \left( A \right) = P\left( {x_t \in \left. A \right|\mathcal{G}_t } \right), $$
(0.2)
where A is a Borel subset of S and
$$ \mathcal{G}_t = \mathop \cap \limits_{\varepsilon > 0} \sigma \left( {Y_s ;s \leqslant t + \varepsilon } \right). $$
(0.3)

Keywords

Covariance Radon Dition Tempo Nite 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    M. Fujisaki, G. Kallianpur and H. Kunita, Stochastic differential equations for the nonlinear filtering problem, Osaka J. Math. 9 (1972), pp. 19–40.MathSciNetMATHGoogle Scholar
  2. [2]
    R.Z. Hashiminsky, Ergodic properties of recurrent diffusion processes and stabilization of the solution of the Cauchy problem for parabolic equations, Theor. Prob. Appl. 5 (1960), pp. 179–196.CrossRefGoogle Scholar
  3. [3]
    K. Ichihara and H. Kunita, A classification of the second order degenerate elliptic operators and its probabilistic characterization, Z. Wahrscheinlichkeitstheorie verw. Gebiete 30 (1974), pp. 235–254.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    R.E. Kaiman and R.S. Bucy, New results in linear filtering and prediction theory, Trans. ASME Ser. D, J. Basic Eng. 83 (1961), pp. 95–108.CrossRefGoogle Scholar
  5. [5]
    H. Kunita, Asymptotic behavior of the nonlinear filtering errors of Markov processes, J. Multivariate Analysis 1 (1971), pp. 365–393.MathSciNetCrossRefGoogle Scholar
  6. [6]
    H. Kunita, Stochastic partial differential equations connected with non-linear filtering, “Nonlinear filtering and stochastic control,” Lect. Notes Math., 1982, pp. 100–169.Google Scholar
  7. [7]
    H. Kunita, Stochastic Flows and Stochastic Differential Equations, in preparation.Google Scholar
  8. [8]
    T.G. Kurtz and D.L. Ocone, Unique characterization of conditional distributions in nonlinear filtering, The Annals of Probabilitiy 16 (1988), pp. 80–107.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    R. Sh. Liptser and A.N. Shiryaev, “Statistics of Random Processes I,” Springer Verlag, New York, 1977.MATHGoogle Scholar
  10. [10]
    P.A. Meyer, “Probability and Potentials,” Blaisdel, Waltham, Massachusetts, 1966.MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • Hiroshi Kunita
    • 1
  1. 1.Department of Applied ScienceKyushu University 36FukuokaJapan

Personalised recommendations