Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Existence of strong solutions for Itô's stochastic equations via approximations
Download PDF
Download PDF
  • Published: June 1996

Existence of strong solutions for Itô's stochastic equations via approximations

  • István Gyöngy1 &
  • Nicolai Krylov2 

Probability Theory and Related Fields volume 105, pages 143–158 (1996)Cite this article

  • 1485 Accesses

  • 233 Citations

  • Metrics details

Summary

Given strong uniqueness for an Itô's stochastic equation with discontinuous coefficients, we prove that its solution can be constructed on “any” probability space by using, for example, Euler's polygonal approximations. Stochastic equations in ℝd and in domains in ℝd are considered.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Alyushina, L.A.: Euler polygonal lines for Itô's equations with monotone coefficients Theory Probab. Appl.32, 340–346 (1987)

    Google Scholar 

  2. Fabes, E.B., Kenig, C.E.: Examples of singular parabolic measures and singular transition probability densities. Duke Mathematical Journal48, 848–856 (1981)

    Google Scholar 

  3. Gyöngy, I., Nualart, D., Sanz-Solé, M.: Approximation and support theorems in modulus spaces. Probab. Theory Relat. Fields101, 495–509 (1995)

    Google Scholar 

  4. Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. Amsterdam Oxford New York: North-Holland 1981

    Google Scholar 

  5. Kaneko, H., Nakao, S.: A note on approximation for stochastic differential equations. Séminaire de Probabilités XXII (Lecture Notes in Mathematics.1321 155–162) Berlin, Heidelberg: Springer, 1988

    Google Scholar 

  6. Krylov, N.V.: On Itô's stochastic integral equations. Theory Probab. Appl.14, 330–336 (1969)

    Google Scholar 

  7. Krylov, N.V.: A simple proof of the existence of a solution of Itô's equation with monotone coefficients. Theory Probab. Appl.3, 583–587 (1990)

    Google Scholar 

  8. Krylov, N.V.: Extremal properties of solutions of stochastic equations. Theory Probab. Appl.29, 205–214 (1984)

    Google Scholar 

  9. Krylov, N.V.: Controlled diffusion processes. New York, Heidelberg, Berlin: Springer, 1980

    Google Scholar 

  10. Maruyama, G.: Continuous Markov processes and stochastic equations. Rend Circ. Mat. Palermo4, 48–90 (1955)

    Google Scholar 

  11. Nakao, S.: On the pathwise uniqueness of solutions of one-dimensional stochastic differential equations. Osaka J. Math.9, 513–518 (1972)

    Google Scholar 

  12. Safonov, M.V.: An example of a diffusion process with singular distribution at some given time. Abstract of Communications, Third Vilnius Conference on Probability Theory and Mathematical Statistics, pp. 133–134, Vilnius, June 22–27, 1981

  13. Stroock, D.W., Vardhan, S.R.S.: Multidimensional diffusion processes. New York: Springer, 1979

    Google Scholar 

  14. Skorokhod, A.V.: Studies in the theory of random processes. New York: Dover, 1982

    Google Scholar 

  15. Veretennikov, A.Yu.: On strong solution and explicit formulas for solutions of stochastic integral equations. Math. USSR Sb.39, 387–403 (1981)

    Google Scholar 

  16. Yamada, T., Watanabe, S.: On the uniqueness of solutions of stochastic differential equations, I, II. J. Math. Kyoto Univ.11, 155–167, 553–563 (1971)

    Google Scholar 

  17. Zvonkin, A.K., Krylov, N.V.: On strong solutions of stochastic differential equations. Sel. Math. Sov.1, 19–61 (1981)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Probability and Statistics, Eötvös Loránd University Budapest, Múzeum krt. 6-8, H-1088, Budapest, Hungary

    István Gyöngy

  2. School of Mathematics, University of Minnesota, 55455, Minneapolis, MN, USA

    Nicolai Krylov

Authors
  1. István Gyöngy
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nicolai Krylov
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Research supported by the Hungarian National Foundation of Scientific Research No. 2990.

Supported in part by NSF Grant DMS-9302516

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Gyöngy, I., Krylov, N. Existence of strong solutions for Itô's stochastic equations via approximations. Probab. Th. Rel. Fields 105, 143–158 (1996). https://doi.org/10.1007/BF01203833

Download citation

  • Received: 06 January 1994

  • Revised: 07 November 1995

  • Issue Date: June 1996

  • DOI: https://doi.org/10.1007/BF01203833

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Mathematics Subject Classification (1991)

  • 60H10
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature