Journal of Theoretical Probability

, Volume 14, Issue 1, pp 125–164 | Cite as

Weak Solutions for SPDEs and Backward Doubly Stochastic Differential Equations

  • V. Bally
  • A. Matoussi


We give the probabilistic interpretation of the solutions in Sobolev spaces of parabolic semilinear stochastic PDEs in terms of Backward Doubly Stochastic Differential Equations. This is a generalization of the Feynman–Kac formula. We also discuss linear stochastic PDEs in which the terminal value and the coefficients are distributions.

stochastic partial differential equation Backward Doubly SDE Feynman–Kac's formula stochastic flows Schwartz distributions weighted Sobolev spaces 


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  1. [A]
    Adams, R. A. (1975). Sobolev Spaces, Academic Press, New York.Google Scholar
  2. [BL]
    Barles, G. and Lesigne, E. (1997). SDE, BSDE and PDE. Pitman Research Notes in Mathematics Series 364, 47–80.Google Scholar
  3. [B]
    Bismut, J. M. (1980). Mécanique Aléatoire. École d'étéde Probabilitéde Saint Flour, 1980. Lect. Notes Math. 929, 5–100.Google Scholar
  4. [EPQ]
    El Karoui, N., Peng, S., and Quenez, M. C. (1997). Backward stochastic differential equations in finance. J. Math. Finance 7(1), 1–71.Google Scholar
  5. [IW]
    Ikeda, N., and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes, 2nd ed., North-HollandûKodansha.Google Scholar
  6. [K1]
    Kunita, H. (1982). Stochastic differential equations and stochastic flows of diffeomorphisms. École d'étéde Probabilitéde Saint-Flour, 1982. Lect. Notes Math. 1097, 143–303.Google Scholar
  7. [K2]
    Kunita, H. (1994). Stochastic flow acting on Schwartz distributions. J. Theor. Prob. 7(2), 247–278.Google Scholar
  8. [K3]
    Kunita, H. (1994). Generalized solutions of a stochastic partial differential equation. J. Theor. Prob. 7(2), 279–308.Google Scholar
  9. [K4]
    Kunita, H. (1990). Stochastic Flows and Stochastic Differential Equations, Cambridge University Press.Google Scholar
  10. [MP]
    Métivier, M., and Pellaumail J. (1980). Stochastic Integration, Academic Press, New York.Google Scholar
  11. [MNS]
    Millet, A., Nualart, D., and Sanz, M., (1989). Integration by parts and time reversal for diffusion processes. The Annals of Probability 17(1), 208–238.Google Scholar
  12. [PP1]
    Pardoux, E., and Peng, S. (1990). Adapted solution of a backward stochastic differential equation. Systems and Control Letters 14, 55–61.Google Scholar
  13. [PP2]
    Pardoux, E., and Peng, S. (1992). Backward SDEs and Quasilinear PDEs. In Rozovskii, B. L., and Sowers, R. B. (eds.), Stochastic Partial Differential Equations and Their Applications, LNCIS 176, Springer.Google Scholar
  14. [PP3]
    Pardoux, E., and Peng, S. (1994). Backward doubly SDEs and systems of quasilinear SPDEs. Probab. Theory Relat. Field 98, 209–227.Google Scholar
  15. [Pe]
    Peng, S. (1993). A nonlinear Feynman_Kac formula and applications. In Chen, S. P., and Young, J. M. (eds), Proc. of Symposium on System Science and Control Theory, pp. 173–184, Singapore, World Scientific, 1992.Google Scholar
  16. [Pr]
    Protter, P. (1995). Stochastic: Integration and differential equations, a new approach. Applications of Mathematics, Springer-Verlag.Google Scholar

Copyright information

© Plenum Publishing Corporation 2001

Authors and Affiliations

  • V. Bally
    • 1
  • A. Matoussi
    • 2
  1. 1.Laboratoire de probabilités, Tour 56Université Paris VIParis Cedex 05France
  2. 2.Laboratoire Statistique et ProcessusUniversité du MaineLe Mans cedexFrance

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