Advertisement

Filtration and Prediction Problems for Stochastic Fields

  • Pavel S. Knopov
  • Olena N. Deriyeva
Chapter
  • 1.2k Downloads
Part of the Springer Optimization and Its Applications book series (SOIA, volume 83)

Abstract

In this chapter we investigate different models of filtration and prediction for stochastic fields generated by some stochastic differential equations. We derive stochastic integro-differentiation equations for an optimal in the mean square sense filter. We also suggest different approaches for finding the best linear estimate for a stochastic field basing on its observations in certain domain. Besides, we investigate the duality of the filtration problem and a certain optimal control problem. This chapter is based on the results published in [3, 5, 11, 12, 17, 41–44, 46, 69].

Keywords

Stochastic Field Filtering Problem Sense Filter Stochastic Parabolic Equations Orthogonal Stochastic Measure 
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.

References

  1. 1.
    Achiezer, N.Y., Glasman, I.M.: Theory of linear operators in hilbert space. Nauka, Мoscow (1966)Google Scholar
  2. 2.
    Bateman, H., Erdelyi, A.: Higher transсendental function. Graw Hill Book Company, New York–Toronto–London (1953)Google Scholar
  3. 3.
    Bazenov, L.G., Knopov, P.S.: On the forecast and filtration problems for random fields satisfying stochastic differential equations. News. Ac. Sci. USSR. Tech. Cybern. 6, 53–157 (1974). In RussianGoogle Scholar
  4. 5.
    Billingsley, P.: Convergence of probability measures. Wiley, New York (1968)zbMATHGoogle Scholar
  5. 11.
    Derieva, E.N.: Stochastic equations of optimal nonlinear filtering of random fields. Cybernetics and system analysis 30(5), 718–725 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 12.
    Derieva, Е.N.: Absolutely continuous change of measure in stochastic differential equations. Cybern Syst Anal 30(6), 943–946 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 15.
    Derieva O.M.: Some properties of measures corresponding to random fields of diffusion type on the plane. Theor. Probab. Math. Stat. 50, 71–76Google Scholar
  8. 16.
    Dorogovtzev A.Ya.: Remarks on the random processes generated by some differential equations. RAS. USSR. (8), 1008–1010 (1962)Google Scholar
  9. 17.
    Dorogovtzev A.Ya., Knopov P.S.: Asymptotic properties of one parametrical estimate for a two-parameter function. Theor. Toch. Proc. (5), 27–35 (1977) (In Russian)Google Scholar
  10. 18.
    Ermoliev, Y.M., Gulenko, V.P., Tsarenko, T.I.: Finite difference approximation methods in optimal control theory. Naukova Dumka, Kiev (1978)Google Scholar
  11. 22.
    Gihman, I.I.: To theory of bi-martingales. Rep. Ac. Sci. Uk. SSR. 6, 9–12 (1982) (In Russian)Google Scholar
  12. 28.
    Gihman, I.I., Skorochod, А.V.: The theory of stochastic processes, vol. 1. Springer, New York (1979)CrossRefzbMATHGoogle Scholar
  13. 35.
    Gushchin, А.А.: On the general theory of random fields on the plane. Russian Math. Surveys. 37(6), 55–80 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 37.
    Kaliath T.: A note on least squares estimation by the innovations method. SIAM J. Control. 10(3) (1972)Google Scholar
  15. 40.
    Knopov P.S.: On one control problem for markov field satisfying a stochastic differential equation. Theor. Optimal. Solut. (3), 109–115 (1969) (In Russian)Google Scholar
  16. 41.
    Knopov P.S.: On filtration and forecast problems for stochastic fields. Cybern. (6), 74–78 (1972) (In Russian)Google Scholar
  17. 42.
    Knopov, P.S.: Optimal estimators of parameters of stochastic system. Naukova Dumka, Kiev (1981). In RussianGoogle Scholar
  18. 43.
    Knopov, P.S.: On an estimator for the drift parameter of random fields. Theor. Probab. Math. Stat. 45, 29–33 (1992)MathSciNetGoogle Scholar
  19. 44.
    Knopov, P.S.: Some applied problems from random field theory. Cybern. Syst. Anal. 46(1), 62–71 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 46.
    Knopov, P.S., Derieva, E.N.: Control problem for diffusion-type random fields. Cybern. Syst. Anal. 31(1), 52–64 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 54.
    Liptser, R.S., Shiriaev, A.N.: Statistics of random processes: General theory. Springer, New York (2001)CrossRefGoogle Scholar
  22. 64.
    Piacetzkaya T.E.: Stochastic integration on the plane and one class of stochastic systems of hyperbolic differential equations. Thesis Ph.D.in Math, Donetzk (1977) (In Russian)Google Scholar
  23. 68.
    Shurko, G.K.: On the convergence of one class of two-parameter stochastic integral equations. Theor. Stoch. Proc. 16, 97–105 (1988). In RussianMathSciNetzbMATHGoogle Scholar
  24. 69.
    Sokolovskiy, V.Z.: Kalman filters for distributed systems (Prepr. 74-60). Institute of Cybernetics, Ukrainian Academy of Sciences Uk SSR, Kiev (1974). In RussianGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Pavel S. Knopov
    • 1
  • Olena N. Deriyeva
    • 1
  1. 1.Department of Mathematical Methods of Operation ResearchV.M. Glushkov Institute of Cybernetics National Academy of Sciences of UkraineKievUkraine

Personalised recommendations