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.
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.
Accepted:
Published online:
Feng Zhang 1
@article{CRMATH_2009__347_9-10_533_0, author = {Feng Zhang}, 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}, }
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/
[1] Backward–forward stochastic differential equations, The Annals of Applied Probability, Volume 3 (1993), pp. 777-793
[2] Weak solutions for SPDEs and backward doubly stochastic differential equations, Journal of Theoretical Probability, Volume 14 (2001), pp. 125-164
[3] SDE, BSDE and PDE, Pitman Research Notes in Mathematics Series, Volume 364 (1997), pp. 47-80
[4] 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] 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] Forward–backward stochastic differential equations and quasilinear parabolic PDEs, PTRF, Volume 114 (1999), pp. 123-150
[7] Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stochastics and Stochastics Reports, Volume 37 (1991), pp. 61-74
[8] 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
Cited by 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).
Comments - Politique