Optimal Linear Nonstationary Filtering

  • Robert S. Liptser
  • Albert N. Shiryaev
Part of the Applications of Mathematics book series (SMAP, volume 5)


On the probability space (Ω, F, P) with a distinguished family of the σ-algebras (F t ), tT, we shall consider the two-dimensional Gaussian random process (θ t , F t ), 0 ≤ tT, satisfying the stochastic differential equations
$$d{\theta _t}\, = \,a(t){\theta _t}dt\, + \,b(t)d{W_1}(t)$$
$$d{\xi _t}\, = \,A(t){\theta _t}dt\, + \,B(t)d{W_2}(t),$$
where W 1 = (W 1(t), F t ) and W 2 = (W 2(t), F t ) are two independent Wiener processes and θ 0, ξ 0 are F 0-measurable.


Conditional Expectation Wiener Process Gaussian Random Process Unique Continuous Solution Skorokhod Space 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


Notes and References. 1

  1. 140.
    Kalman, R.E. and Bucy, R.S. (1961): New results in linear filtering and prediction theory. Trans. ASME, 83D, 95–100MathSciNetCrossRefGoogle Scholar
  2. 296.
    Stratonovich, R.L. (1966): Conditional Markov Processes and their Applications to Optimal Control Theory. Izd. MGU, MoscowGoogle Scholar
  3. 270.
    Ruymgaart, P.A. (1971): A note on the integral representation of the KalmanBucy estimate. Indag. Math., 33, 4, 346–60.MathSciNetGoogle Scholar
  4. 214.
    Liptser, R.S. and Shiryaev, A.N. (1989): Theory of Martingales. Kluwer, Dordrecht (Russian edition 1986 )zbMATHCrossRefGoogle Scholar

Notes and References.2

  1. 38.
    Chow, P.L., Khasminskii, R.Z. and Liptser, R.S. (1997): Tracking of a signal and its derivatives in Gaussian white noise. Stochastic Processes Appl., 69, 2, 259–73MathSciNetzbMATHCrossRefGoogle Scholar
  2. 240.
    Miller, B.M. and Runggaldier, W.J. (1997) Kalman filtering for linear systems with coefficients driven by a hidden Markov jump process. Syst. Control Lett., 31, 93–102MathSciNetzbMATHCrossRefGoogle Scholar
  3. 238.
    Miller, B.M. and Rubinovich, E.Ya. (1995): Regularization of a generalized Kalman filter. Math. Comput. Simul., 39, 87–108MathSciNetCrossRefGoogle 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

Personalised recommendations