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
Linear stochastic differential equations with boundary conditions
Download PDF
Download PDF
  • Published: August 1989

Linear stochastic differential equations with boundary conditions

  • Daniel Ocone1 &
  • Etienne Pardoux2 

Probability Theory and Related Fields volume 82, pages 489–526 (1989)Cite this article

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Summary

We study linear stochastic differential equations with affine boundary conditions. The equation is linear in the sense that both the drift and the diffusion coefficient are affine functions of the solution. The solution is not adapted to the driving Brownian motion, and we use the extended stochastic calculus of Nualart and Pardoux [16] to analyse them. We give analytical necessary and sufficient conditions for existence and uniqueness of a solution, we establish sufficient conditions for the existence of probability densities using both the Malliavin calculus and the co-aera formula, and give sufficient conditions that the solution be either a Markov process or a Markov field.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Adams, M.B., Willsky, A.S., Levy, B.C.: Linear estimation of boundary value stochastic processes. IEEE Trans. Autom. Control, AC-29, 803–821 (1984)

    Google Scholar 

  2. Bismut, J-M.: Martingales, the Malliavin calculus and hypoellipticity under general Hörmander's conditions. Z. Wahrscheinlichkeitstheor. Verw. Geb. 56, 469–505 (1981)

    Google Scholar 

  3. Bouleau, N., Hirsch, F.: Propriétés d'absolue continuité dans les espaces de Dirichlet et applications aux équations differentielles, in Séminaire de Probabilités XX. (Lect. Notes Math., vol. 1204) Berlin Heidelberg New York: Springer 1986

    Google Scholar 

  4. Chay, S.C.: On quasi-Markov random fields. J. Multivariate Anal. 2, 14–76 (1972)

    Google Scholar 

  5. Cinlar, E., Wang, J.G.: Random circles and fields on circles. Preprint

  6. Ethier, S.N., Kurtz, T.G.: Markov processes; characterization and convergence. New York: Wiley 1986

    Google Scholar 

  7. Federer, H.: Geometric Measure Theory. Berlin Heidelberg New York: Springer 1969

    Google Scholar 

  8. Helgason, S.: Differential geometry, Lie groups, and symmetric spaces. New York: Academic Press 1978

    Google Scholar 

  9. Jamison, B.: Reciprocal processes, the stationary gaussian case. Ann. Math. Stat. 41, 1624–1630 (1970)

    Google Scholar 

  10. Jamison, B.: Reciprocal processes. Z. Wahrscheinlichkeitstheor. Verw. Geb. 30, 65–86 (1974)

    Google Scholar 

  11. Jeulin, T.: Semi-martingales et grossissement d'une filtration. (Lect. Notes Math., vol. 833) Berlin Heidelberg New York: Springer 1980

    Google Scholar 

  12. Jeulin, T., Yor, M. (eds.): Grossissement de filtrations; exemples et applications. Lect. Notes Math. 1118. Berlin Heidelberg New York: Springer 1985

    Google Scholar 

  13. Krener, A.: Reciprocal processes and the stochastic realization problem for acausal systems, in C.I. Byrnes and A. Lindquist, eds., Modelling, identification, and robust control. Amsterdam New York: Elsevier-North Holland 1986

    Google Scholar 

  14. Kwakernaak, H.: Periodic linear differential stochastic processes. SIAM J. Control Optimization 13, 400–413 (1975)

    Google Scholar 

  15. Meyer, P.-A.: Un cours sur les intégrales stochastiques Séminaire de Probabilités X. (Lect. Notes in Math., vol. 511) Berlin Heidelberg New York: Springer 1976

    Google Scholar 

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

    Google Scholar 

  17. Ocone, D., Párdoux, E.: A generalized Itô-Ventzell formula: Application to a class of anticipating stochastic differential equations. Ann. IHP 25, 39–71 (1989)

    Google Scholar 

  18. Russek, A.: Gaussian n-Markovian processes and stochastic boundary value problems. Z. Wahrscheinlichkeitstheor. Verw. Geb. 53, 117–122 (1980)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics Department, Rutgers University, 08903, New Brunswick, NJ, USA

    Daniel Ocone

  2. Mathématiques, UA 225, Université de Provence, F-13331, Marseille Cedex 3, France

    Etienne Pardoux

Authors
  1. Daniel Ocone
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Etienne Pardoux
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Supported in part by NSF Grant No. MCS-8301880

The research was carried out while this author was visiting the Institute for Advanced Study, Princeton NJ, and was supported by a grant from the RCA corporation

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Ocone, D., Pardoux, E. Linear stochastic differential equations with boundary conditions. Probab. Th. Rel. Fields 82, 489–526 (1989). https://doi.org/10.1007/BF00341281

Download citation

  • Received: 30 November 1987

  • Issue Date: August 1989

  • DOI: https://doi.org/10.1007/BF00341281

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

Keywords

  • Boundary Condition
  • Differential Equation
  • Diffusion Coefficient
  • Probability Density
  • Stochastic Process
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