Comptes Rendus
no. 9-10
Partial Differential Equations/Probability Theory
Sobolev weak solutions for parabolic PDEs and FBSDEs
[Solutions faibles de Sobolev des EDP paraboliques et des EDSR]

Présenté par : Pierre-Louis Lions

Feng Zhang1
1 School of Mathematics, Shandong University, Jinan, 250100, China
Comptes Rendus. Mathématique, Volume 347 (2009) no. 9-10, pp. 533-536.

Cette Note est consacré à la représentation des solutions faibles au sens de Sobolev d'une EDP quasi-linéaire parabolique avec des coefficients monotones par un système d'EDS et d'EDSR. Un caractère distinctif de ce résultat est que la composante de l'EDS est couplée avec les solutions d'EDSR.

This Note is devoted to the representation of Sobolev weak solutions to quasi-linear parabolic PDEs with monotone coefficients via FBSDEs. One distinctive character of this result is that the forward component of the FBSDE is coupled with the backward variable.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2009.03.002
Feng Zhang 1

1 School of Mathematics, Shandong University, Jinan, 250100, China 
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Feng Zhang. Sobolev weak solutions for parabolic PDEs and FBSDEs. Comptes Rendus. Mathématique, Volume 347 (2009) no. 9-10, pp. 533-536. doi : 10.1016/j.crma.2009.03.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.03.002/

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[2] V. Bally; A. Matoussi Weak solutions for SPDEs and backward doubly stochastic differential equations, Journal of Theoretical Probability, Volume 14 (2001), pp. 125-164

[3] G. Barles; E. Lesigne SDE, BSDE and PDE, Pitman Research Notes in Mathematics Series, Volume 364 (1997), pp. 47-80

[4] Y. Ouknine; I. Turpin Weak solutions of semilinear PDEs in Sobolev spaces and their probabilistic interpretation via the FBSDEs, Stochastic Analysis and Applications, Volume 24 (2006), pp. 871-888

[5] E. Pardoux; S. Peng Backward stochastic differential equations and quasilinear parabolic partial differential equations (B.L. Rozoviskii; R.B. Sowers, eds.), Stochastic Partial Differential Equations and Their Applications, LNCIS, vol. 176, Springer, 1992, pp. 200-217

[6] E. Pardoux; S. Tang Forward–backward stochastic differential equations and quasilinear parabolic PDEs, PTRF, Volume 114 (1999), pp. 123-150

[7] S. Peng Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stochastics and Stochastics Reports, Volume 37 (1991), pp. 61-74

[8] Z. Wu; Z. Yu Dynamic programming principle for one kind of stochastic recursive optimal control problem and Hamilton–Jacobi–Bellman equations, SIAM J. Control and Optimization, Volume 47 (2008), pp. 2616-2641

Cité par Sources :

☆ This work was supported by the National Natural Science Foundation (10671112), the National Basic Research Program of China (973 Program, No. 2007CB814904), the Natural Science Foundation of Shandong Province (Z2006A01) and the Doctoral Fund of Education Ministry of China (20060422018).

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