Advertisement

Ukrainian Mathematical Journal

, Volume 57, Issue 9, pp 1424–1441 | Cite as

Smoothing Problem in Anticipating Scenario

  • A. A. Dorogovtsev
Article

Abstract

We consider a smoothing problem for stochastic processes satisfying stochastic differential equations with Wiener processes that may not have a semimartingale property with respect to the joint filtration.

Keywords

Differential Equation Stochastic Process Stochastic Differential Equation Wiener Process Smoothing Problem 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

REFERENCES

  1. 1.
    K. Ito and H. P. McKean, Jr., “Diffusion processes and their sample paths. Second printing, corrected,” in: Grundlehren Math. Wissenschaften, Springer, Berlin-New York (1974).Google Scholar
  2. 2.
    B. Simon, The P(φ) 2 Euclidean (Quantum) Field Theory, Princeton Univ. Press (1974).Google Scholar
  3. 3.
    A. A. Dorogovtsev, Stochastic Analysis and Random Maps in Hilbert Space, VSP, Utrecht-Tokyo (1994).Google Scholar
  4. 4.
    A. A. Dorogovtsev, “Anticipating equations and filtration problem,” Theory Stochast. Processes, 3(19), Issue 1–2, 154–163 (1997).zbMATHGoogle Scholar
  5. 5.
    A. A. Dorogovtsev, “Conditional measures for diffusion processes and anticipating stochastic equations, ” Theory Stochast. Processes, 4(20), 17–24 (1998).zbMATHMathSciNetGoogle Scholar
  6. 6.
    A. V. Skorokhod, “One generalization of the stochastic integral,” Probab. Theory Its Appl., 20, No.2, 223–237 (1975).zbMATHGoogle Scholar
  7. 7.
    R. Sh. Liptser and A. N. Shyriaev, Statistics of Random Processes [in Russian], Nauka, Moscow (1974).Google Scholar
  8. 8.
    P. Billingsley, Convergence of Probability Measures, Wiley, New York (1968).Google Scholar
  9. 9.
    B. B. Mandelbrot and J. van Ness, “Fractional Brownian motion, fractional noises and applications, ” SIAM Rev., 10, 422–437 (1968).CrossRefMathSciNetGoogle Scholar
  10. 10.
    S. Tindel, C. A. Tudor, and F. Viens, “Stochastic evolution equations with fractional Brownian motion, ” Probab. Theory Relat. Fields, 127, No.2, 186–204 (2003).CrossRefMathSciNetGoogle Scholar
  11. 11.
    A. A. Dorogovtsev, “An action of Gaussian strong random operator on random elements,” Probab. Theory Its Appl., 31, No.4, 811–814 (1986).zbMATHMathSciNetGoogle Scholar
  12. 12.
    P. Malliavin, “Stochastic analysis,” Text. Monograph, Grundlehren Math. Wissenschaften, 313, Springer, Berlin (1997).Google Scholar
  13. 13.
    D. Nualart, “The Malliavin calculus and related topics,” Text. Monograph, Probability and its Application, Springer, New York (1995).Google Scholar
  14. 14.
    A. A. Dorogovtsev, “Stochastic integration and one class of Gaussian stochastic processes,” Ukr. Mat. Zh., 50, No.4, 495–505 (1998).zbMATHMathSciNetGoogle Scholar
  15. 15.
    A. S. Ustunel and M. Zakai, Transformation of Measure on Wiener Space, Springer (2000).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • A. A. Dorogovtsev
    • 1
  1. 1.Institute of MathematicsUkrainian Academy of SciencesKievUkraine

Personalised recommendations