Conditionally Gaussian Processes

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


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
$${{\pi }_{t}}\left( {{{\theta }^{2}}} \right) = M\left( {\theta _{t}^{2}\left| {\mathcal{F}_{t}^{\xi }} \right.} \right),{{\pi }_{t}} = \left( {{{\theta }^{3}}} \right) = M\left( {\theta _{t}^{3}\left| {\mathcal{F}_{t}^{\xi }} \right.} \right)$$


Random Process Gaussian Process Conditional Distribution Extended Kalman Filter Wiener Process 
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    Liptser, R.S. (1967): On filtering and extrapolation of the components of diffusion type Markov processes. Teor. Veroyatn. Primen., 12, 4, 754–6MathSciNetGoogle Scholar
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    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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