Optimal Filtering, Interpolation and Extrapolation of Markov Processes with a Countable Number of States

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


The present chapter will be concerned with a pair of random processes (θ, ξ) = (θ t , ξ t ), 0 ≤ tT, where the unobservable component θ is a Markov process with a finite or countable number of states, and the observable process ξ permits the stochastic differential
$$d{\xi _t}\,{A_t}({\theta _t},\xi )dt\, + \,{B_t}(\xi )d{W_t},$$
where W t is a Wiener process.


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Notes and References. 1

  1. 210.
    Liptser, R.S. and Shiryaev, A.N. (1969): Interpolation and filtering of the jump component of a Markov process. Izv. Akad. Nauk SSSR, Ser. Mat., 33, 4, 901–14MathSciNetzbMATHGoogle Scholar
  2. 267.
    Rozovskii, B.L. and Shiryaev, A.N. (1972): On infinite systems of stochastic differential equations arising in the theory of optimal nonlinear filtering. Teor. Veroyatn. Primen., 17, 2, 228–37Google Scholar
  3. 279.
    Shiryaev, A.N. (1966): Stochastic equations of nonlinear filtering of jump Markov processes. Probi. Peredachi Inf., 2, 3, 3–22Google Scholar
  4. 296.
    Stratonovich, R.L. (1966): Conditional Markov Processes and their Applications to Optimal Control Theory. Izd. MGU, MoscowGoogle Scholar
  5. 312.
    Wonham, W.M. (1965): Some applications of stochastic differential equations to optimal nonlinear filtering. SIAM J. Control Optimization, 2, 347–69MathSciNetzbMATHGoogle Scholar

Notes and References. 2

  1. 60.
    Elliott, R.J., Aggoun, L. and Moore, J.B. (1995): Hidden Markov Models. Springer-Verlag, New York Berl in HeidelbergzbMATHGoogle Scholar
  2. 168.
    Kunita, H. (1971): Ergodic properties of nonlinear filtering processes. In: Spatial Stochastic Processes. K. Alexander and J. Watkins (eds). Birkhäuser, Boston, 233–56Google 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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