Probability Theory and Related Fields

, Volume 98, Issue 2, pp 209–227 | Cite as

Backward doubly stochastic differential equations and systems of quasilinear SPDEs

  • Etienne Pardoux
  • Shige Peng


We introduce a new class of backward stochastic differential equations, which allows us to produce a probabilistic representation of certain quasilinear stochastic partial differential equations, thus extending the Feynman-Kac formula for linear SPDE's.

Mathematics Subject Classification

60H10 60H15 60H30 


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  1. 1.
    Krylov, N.V., Rozovskii, B.L.: Stochastic evolution equations. J. Sov. Math.16, 1233–1277 (1981)Google Scholar
  2. 2.
    Nualart, D., Pardoux, E.: Stochastic calculus with anticipating integrands. Probab. Theory Relat. Fields78, 535–581 (1988)Google Scholar
  3. 3.
    Ocone, D., Pardoux, E.: A stochastic Feynman-Kac formula for anticipating SPDEs, and application to nonlinear smoothing. Stochastics45, 79–126 (1993)Google Scholar
  4. 4.
    Pardoux, E.: Un résultat sur les équations aux dérivés partielles stochastiques et filtrage des processus de diffusion. Note C.R. Acad. Sci., Paris Sér. A287, 1065–1068 (1978)Google Scholar
  5. 5.
    Pardoux, E.: Stochastic PDEs, and filtering of diffusion processes. Stochastics3, 127–167 (1979)Google Scholar
  6. 6.
    Pardoux, E., Peng, S.: Adapted solution of a backward stochastic differential equation. Syst. Control Lett.14, 55–61 (1990)Google Scholar
  7. 7.
    Pardoux, E., Peng, S.: Backward stochastic differential equations and quasilinear parabolic partial differential equations. In: Rozuvskii, B.L., Sowers, R.B. (eds.) Stochastic partial differential equations and their applications. (Lect. Notes Control Inf. Sci., vol. 176, pp. 200–217) Berlin Heidelberg New York: Springer 1992Google Scholar
  8. 8.
    Peng, S.: Probabilistic interpretation for systems of quasilinear parabolic partial differential equations. Stochastics37, 61–74 (1991)Google Scholar
  9. 9.
    Peng, S.: A generalized dynamic programming principle and Hamilton-Jacobi-Bellmann equation. Stochastics38, 119–134 (1992)Google Scholar
  10. 10.
    Peng, S.: A non linear Feynman-Kac formula and applications. In: Chen, S.P., Yong, J.M. (eds.) Proc. of Symposium on system science and control theory, pp. 173–184. Singapore: World Scientific 1992Google Scholar
  11. 11.
    Rozovskii, B.: Stochastic evolution systems. Dordrecht: Reidel 1991Google Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • Etienne Pardoux
    • 1
  • Shige Peng
    • 2
  1. 1.Laboratoire APT, URA 225Université de ProvenceMarseille Cedex 3France
  2. 2.Institute of MathematicsShandong UniversityJinanPeople's Republic of China

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