Stochastic Differential Equations on the Plane

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


In this chapter we investigate diffusion-type fields and Ito fields on the plane, two-parameter version of the Girsanov theorem, weak and strong solutions of stochastic differential equations on the plane, and the probability measures generated by stochastic fields. The results presented in this chapter are published in [10, 12, 14, 16, 25, 42, 44, 45, 47, 48, 65, 71].


Stochastic Differential Equations Stochastic Field Girsanov Theorem Strong Solution Wiener Field 
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.


  1. 4.
    Benes, V.E.: Existence of optimal stochastic control laws. SIAM J. Control 9(3), C446–C472 (1971)MathSciNetCrossRefGoogle Scholar
  2. 6.
    Brossard, J., Chevalier, L.: Calcul stochastique et inequalites de norm pour les martingales bi-browniennes. Application aux functions bi-harmoniques. Ann. Inst. Fourier. 30(4), 97–120 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 10.
    Derieva, E.N.: Generalized girsanov theorem for two parameter martingales. Control and decision making methods under risks and uncertainty, pp. 43–50. V.M.Glushkov Institute of Cybernetics, Kiev (1993). In RussianGoogle Scholar
  4. 12.
    Derieva, Е.N.: Absolutely continuous change of measure in stochastic differential equations. Cybern Syst Anal 30(6), 943–946 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 13.
    Deriyeva O.M.: Control and identification problems for diffusion-type processes with distributed parameters. The international conference on Hanshan. Chernivtzi, Ruta, p. 39 (1994) (In Ukrainian)Google Scholar
  6. 14.
    Deriyeva O.M.: On some properties of measures corresponding to diffusion-type random fields on the plane. Krainian conference of Yang scientists. Mathematics, Kiev University, Kiev, 20 July 1994, pp. 253–260 (1994) (In Ukrainian)Google 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. 23.
    Gihman, I.I.: Two-parameter martingales. Russian Math. Surveys. 37(6), 1–30 (1982)CrossRefGoogle Scholar
  11. 25.
    Gihman I.I., Pyasetskaya T.E.: On a class of stochastic prtial differential equations containing two-parametric white noise. limit theor for random process. Akad. Nauk Ukr. SSR. Inst. Mat. 71–92 (1977) (In Russian)Google Scholar
  12. 42.
    Knopov, P.S.: Optimal estimators of parameters of stochastic system. Naukova Dumka, Kiev (1981). In RussianGoogle Scholar
  13. 43.
    Knopov, P.S.: On an estimator for the drift parameter of random fields. Theor. Probab. Math. Stat. 45, 29–33 (1992)MathSciNetGoogle Scholar
  14. 44.
    Knopov, P.S.: Some applied problems from random field theory. Cybern. Syst. Anal. 46(1), 62–71 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 45.
    Knopov, P.S., Derieva, O.M.: A generalized Girsanov theorem and its statistical applications. Theor. Probab. Math. Stat. 51, 69–79 (1995)MathSciNetGoogle Scholar
  16. 47.
    Knopov, P.S., Statland, Е.S.: On absolutely continuous measures corresponding to some stochastic fields on the plane. Statistic and control problems for random processes, pp. 158–169. Institute of Cybernetics, Ukrainian Academy of Sciences Uk SSR, Kiev (1973). In RussianGoogle Scholar
  17. 48.
    Knopov P.S., Statland Е.S.: On hypothesis distinсtion for some classes of stochastic system with distributed parameters. Cybern. (2), C106–C107 (1974) (In Russian)Google Scholar
  18. 52.
    Ledoux, M.: Inegalites de Burkholder pour martingales indexes par N×N. Lect. Notes. Math. 863, 122–127 (1981)MathSciNetCrossRefGoogle Scholar
  19. 54.
    Liptser, R.S., Shiriaev, A.N.: Statistics of random processes: General theory. Springer, New York (2001)CrossRefGoogle Scholar
  20. 64.
    Piacetzkaya T.E.: Stochastic integration on the plane and one class of stochastic systems of hyperbolic differential equations. Thesis Math, Donetzk (1977) (In Russian)Google Scholar
  21. 65.
    Ponomarenko, L.L.: Linear filtering of random fields controlled by stochastic equations. Cybern. Syst. Anal. 9(2), 287–292 (1973)MathSciNetCrossRefGoogle Scholar
  22. 71.
    Tzarenko, T.I.: The existence and uniqueness of solutions of stochastic Darbou equations. Theory of optimal solutions. Institute of Cybernetics, Kiev (1973). 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