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

Zhang, Feng1
1 School of Mathematics, Shandong University, Jinan, 250100, China
Comptes Rendus. Mathématique, Tome 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
Zhang, Feng 1

1 School of Mathematics, Shandong University, Jinan, 250100, China 
@article{CRMATH_2009__347_9-10_533_0,
     author = {Zhang, Feng},
     title = {Sobolev weak solutions for parabolic {PDEs} and {FBSDEs}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {533--536},
     publisher = {Elsevier},
     volume = {347},
     number = {9-10},
     year = {2009},
     doi = {10.1016/j.crma.2009.03.002},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2009.03.002/}
}
TY  - JOUR
AU  - Zhang, Feng
TI  - Sobolev weak solutions for parabolic PDEs and FBSDEs
JO  - Comptes Rendus. Mathématique
PY  - 2009
SP  - 533
EP  - 536
VL  - 347
IS  - 9-10
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2009.03.002/
DO  - 10.1016/j.crma.2009.03.002
LA  - en
ID  - CRMATH_2009__347_9-10_533_0
ER  - 
%0 Journal Article
%A Zhang, Feng
%T Sobolev weak solutions for parabolic PDEs and FBSDEs
%J Comptes Rendus. Mathématique
%D 2009
%P 533-536
%V 347
%N 9-10
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2009.03.002/
%R 10.1016/j.crma.2009.03.002
%G en
%F CRMATH_2009__347_9-10_533_0
Zhang, Feng. Sobolev weak solutions for parabolic PDEs and FBSDEs. Comptes Rendus. Mathématique, Tome 347 (2009) no. 9-10, pp. 533-536. doi : 10.1016/j.crma.2009.03.002. http://www.numdam.org/articles/10.1016/j.crma.2009.03.002/

[1] Antonelli, F. Backward–forward stochastic differential equations, The Annals of Applied Probability, Volume 3 (1993), pp. 777-793

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

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

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

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

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

[8] Wu, Z.; Yu, Z. 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).