184
Views
9
CrossRef citations to date
0
Altmetric
Original Articles

Weak Solutions of Semilinear PDEs in Sobolev Spaces and Their Probabilistic Interpretation via the FBSDEs

&
Pages 871-888 | Accepted 10 Feb 2006, Published online: 21 Sep 2006

REFERENCES

  • Bally , V. , E. Pardoux , and L. Stoica . 2005 . Backward stochastic differential equations associated to a symetric Markov process . Potential Anal. 22 ( 1 ): 17 – 60 . [CSA]
  • Fuhrman , M. , and G. Tessitore . 2005 . Generalized directional gradients, backward stochastic differential equations and mild solutions of semilinear parabolic equations . Appl. Math. Optim. 51 : 279 – 332 . [CSA] [CROSSREF]
  • Crandall , M.G. , and P.L. Lions . 1983. Viscosity solutions of Hamilton-Jacobi equations. Transactions of the American Mathematical Society 277: 1–42. [CSA] [CROSSREF]
  • Pardoux , E. , and S. Peng . 1992 . Backward SDEs and quasilinear PDEs . In Stochastic Partial Differential Equations and Their Applications . eds. B.L. Rozovskii , and R.B. Sowers . LNCIS 176 . Springer .
  • Barles , G. , and E. Lesigne . 1997 . SDE, BSDE and PDE . Pitman Research Notes in Maths Series, 364 , pp. 47 – 80 . Harlow : Longman .
  • Bally , V. , and A. Matoussi . 2001 . Weak solutions for SPDEs and backward doubly stochastic differential equations . J Theoret Prob. 14 ( 1 ): 125 – 164 . [CSA] [CROSSREF]
  • Kunita , H. 1994 . Stochastic flow acting on Schwartz distributions . J. Theor. Prob. 7 ( 2 ): 247 – 278 . [CSA] [CROSSREF]
  • Kunita , H. 1982 . Stochastic differential equations and stochastic flows of diffeomorphisms . Ecole d'i ti de Probabiliti de Saint-Flour. Ecole d'eté de-Saint-Flour XII. Lect. Notes Math, 1097 . pp. 143 – 303 . [CSA]
  • Pardoux , E. 1998 . Backward stochastic differential equations and viscosity solutions of systems of semilinear parabolic and elliptic PDEs of second order . Stochastic Analysis and Related Topics VI ( Geilo, 1996). Progr. Probab . Vol. 42 . pp. 79 – 127 . Boston , MA : Birkhdauser .
  • Bouleau , N. , and F. Hirsh . 1989 . On the derivability with respect to the initial data of the solution of a stochastic differential equation with Lipschitz coefficients . Lect Notes in Math. 1393 , pp. 39 – 57 . [CSA]
  • N'Zi , M. , Y. Ouknine , and A. Sulem . 2003 . Regularity and representation of viscosity solutions of partial differential equations via backward stochastic differential equations . Stoch. Proc. [CSA]
  • Matoussi , A. , 1998 . Equations Differentielles Stochastiques ri trogrades Ri ichies Coefficients Continus, Solutions faibles d'EDPS et d'EDDSR . Thhse de Doctorat de l'universiti du Maine .
  • Bouleau , N. , and F. Hirsh . 1986 . Formes de Dirichlet généralisées et densité des variables aléatoires réelles sur l'espace de Wiener . J. Funct. Anal. 69 : 229 – 259 . [CSA] [CROSSREF]
  • Bouleau , N. , and F. Hirsh . 1991 . Dirichlet Forms and Analysis on Wiener Space . Gruyter Studies in Math. 14 . Berlin-New York .

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.