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Conditionally Gaussian Processes

  • Robert S. Liptser
  • Albert N. Shiryaev
Part of the Stochastic Modelling and Applied Probability book series (SMAP, volume 6)

Abstract

Let (θ, ξ) = (θ t , ξ t ), 0 ≤ tT, be a random process with unobservable first component and observable second component. In employing the equations of optimal nonlinear filtering given by (8.10) one encounters an essential difficulty: in order to find π t (θ), it is necessary to know the conditional moments of the higher orders
$${{\pi }_{t}}\left( {{{\theta }^{2}}} \right) = M\left( {\theta _{t}^{2}\left| {\mathcal{F}_{t}^{\xi }} \right.} \right),{{\pi }_{t}} = \left( {{{\theta }^{3}}} \right) = M\left( {\theta _{t}^{3}\left| {\mathcal{F}_{t}^{\xi }} \right.} \right)$$
.

Keywords

Random Process Gaussian Process Conditional Distribution Extended Kalman Filter Wiener Process 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 194.
    Liptser, R.S. (1967): On filtering and extrapolation of the components of diffusion type Markov processes. Teor. Veroyatn. Primen., 12, 4, 754–6MathSciNetGoogle Scholar
  2. 205.
    Liptser, R.S. and Shiryaev, A.N. (1968): Nonlinear filtering of diffusion type Markov processes. Tr. Mat. Inst. Steklova, 104, 135–80Google Scholar
  3. 253.
    Picard, J. (1991): Efficiency of the extended (Kalman) filter for nonlinear systems with small noise. SIAM J. Appl. Math., 51, 3, 843–85MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Robert S. Liptser
    • 1
  • Albert N. Shiryaev
    • 2
  1. 1.Department of Electrical Engineering SystemsTel Aviv UniversityTel AvivIsrael
  2. 2.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

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