Skip to main content

On the derivability, with respect to the initial data, of the solution of a stochastic differential equation with Lipschitz coefficients

Part of the Lecture Notes in Mathematics book series (LNM,volume 1393)

Abstract

We consider a stochastic differential equation, driven by a Brownian motion, with Lipschitz coefficients. We prove that the corresponding flow is, almost surely, almost everywhere derivable with respect to the initial data for any time, and the process defined by the Jacobian matrices is a GLn(ℝ)-valued continuous solution of a linear stochastic differential equation. In dimension one, this process is given by an explicit formula.

These results partially extend those which are known when the coefficients are C1,α-Hölder continuous. Dirichlet forms are involved in the proofs.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   34.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   46.00
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. N. BOULEAU et F. HIRSCH: Formes de Dirichlet générales et densité des variables aléatoires réelles sur l'espace de Wiener, J.Funct.Anal. 69 (1986), 229–259.

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. N.BOULEAU et F.HIRSCH: Propriétés d'absolue continuité dans les espaces de Dirichlet et application aux équations différentielles stochastiques, in "Séminaire de Probabilités XX, 1984/85", 131–161, Lect. N. in M.1204, Springer (1986).

    Google Scholar 

  3. N. BOULEAU et F. HIRSCH: Sur des propriétés du flot d'une équation différentielle stochastique, C.R.Acad.Sci.Paris, 306 (1988), 421–424.

    MathSciNet  MATH  Google Scholar 

  4. J.DENY: Méthodes hilbertiennes en théorie du potentiel, in C.I.M.E. Potential Theory, Cremonese, 1970.

    Google Scholar 

  5. M. FUKUSHIMA: Dirichlet forms and Markov processes, North-Holland, Amsterdam, 1980.

    MATH  Google Scholar 

  6. H.KUNITA: Stochastic differential equations and stochastic flows of diffeomorphisms, in "Ecole d'Eté de Saint Four XII, 1982", 143–303, Lect. N. in M. 1097, Springer (1984).

    Google Scholar 

  7. A.MILLET, D.NUALART and M.SANZ: Integration by parts and time reversal for diffusions processes, Annals of Proba. (to appear).

    Google Scholar 

  8. D.W. STROOCK: The Malliavin calculus, a functional analytic approach, J.Funct. Anal. 44 (1981), 212–257.

    CrossRef  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1989 Springer-Verlag

About this paper

Cite this paper

Bouleau, N., Hirsch, F. (1989). On the derivability, with respect to the initial data, of the solution of a stochastic differential equation with Lipschitz coefficients. In: Bouleau, N., Feyel, D., Mokobodzki, G., Hirsch, F. (eds) Séminaire de Théorie du Potentiel Paris, No. 9. Lecture Notes in Mathematics, vol 1393. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0085770

Download citation

  • DOI: https://doi.org/10.1007/BFb0085770

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-51592-0

  • Online ISBN: 978-3-540-46675-8

  • eBook Packages: Springer Book Archive