Skip to main content

On some sample path properties of Skorohod integral processes

Part of the Lecture Notes in Mathematics book series (SEMPROBAB,volume 1526)

Abstract

Four examples are presented which show that stochastic integral processes with anticipating integrands can have very different sample path behaviour from those with adapted ones. For example, Skorohod integral processes need not be semimartingales, but can still have smooth occupation densities. Moreover, even if they are continuous, and have finite quadratic variation, this may still be essentially bigger than expected for “smooth” integrands.

Key words and phrases

  • Skorohod integral
  • occupation densities
  • semimartingales
  • quadratic variation
  • AMS 1985 subject classifications
  • primary 60 H 05, 60 G 17
  • secondary 60 H 20, 60 J 65

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   39.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
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. Berman, S.M. Local times and sample function properties of stationary Gaussian processes. Trans. A.M.S. 137 (1969), 277–299.

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. Buckdahn, R. Quasilinear partial stochastic differential equations without nonanticipation requirement. Preprint Nr. 176, Humboldt-Universitaet Berlin (1988).

    Google Scholar 

  3. Buckdahn, R. Transformations on the Wiener space and Skorohod-type stochastic differential equations. Seminarbericht Nr. 105, Sektion Mathematik, Humboldt-Universitaet Berlin (1989).

    Google Scholar 

  4. Buckdahn, R. The nonlinear transformation of the Wiener measure. Preprint Nr. 253, Humboldt-Universitaet Berlin (1990).

    Google Scholar 

  5. Donati-Martin, C. Equations différentielles stochastiques dans R avec conditions aux bords. Preprint, Univ. de Provence (1990).

    Google Scholar 

  6. Imkeller, P. Existence and continuity of occupation densities of stochastic integral processes. Preprint, Universitaet Muenchen (1991).

    Google Scholar 

  7. Jeulin, Th. Semi-martingales et grossissement d'une filtration. LNM 833. Springer: Berlin, Heidelberg, New York (1980).

    CrossRef  MATH  Google Scholar 

  8. Marcus, M.B. Hölder conditions for Gaussian processes with stationary increments. Trans. A.M.S. 134 (1968), 29–52.

    MATH  Google Scholar 

  9. Nualart, D., Pardoux, E. Stochastic calculus with anticipating integrands. Probab. Th. Rel. Fields 78 (1988), 535–581.

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. Nualart, D., Pardoux, E. Boundary value problems for stochastic differential equations. Preprint (1990).

    Google Scholar 

  11. Nualart, D., Pardoux, E. Second order stochastic differential equations with Dirichlet boundary conditions. Preprint (1990).

    Google Scholar 

  12. Ocone, D., Pardoux, E. Linear stochastic differential equations with boundary conditions. Probab. Th. Rel. Fields 82 (1989), 489–526.

    CrossRef  MathSciNet  MATH  Google Scholar 

  13. Ocone, D., Pardoux, E. A generalized Itô-Ventzell formula. Applications to a class of anticipating stochastic differential equations. Ann. Inst. H. Poincaré 25 (1989), 39–71.

    MathSciNet  MATH  Google Scholar 

  14. Pardoux, E., Protter, Ph. A two-sided stochastic integral and its calculus. Probab. Th. Rel. Fields 76 (1987), 15–50.

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. Protter, Ph. Stochastic integration and differential equations. A new approach. Applications of Mathematics. Springer: Berlin, Heidelberg, New York (1990).

    CrossRef  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1992 Springer-Verlag

About this paper

Cite this paper

Barlow, M.T., Imkeller, P. (1992). On some sample path properties of Skorohod integral processes. In: Azéma, J., Yor, M., Meyer, P.A. (eds) Séminaire de Probabilités XXVI. Lecture Notes in Mathematics, vol 1526. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0084311

Download citation

  • DOI: https://doi.org/10.1007/BFb0084311

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-56021-0

  • Online ISBN: 978-3-540-47342-8

  • eBook Packages: Springer Book Archive