We consider a system of fully coupled forward-backward stochastic differential equations. First we generalize the results of Pardoux-Tang [7] concerning the regularity of the solutions with respect to initial conditions. Then, we prove that in some particular cases this system leads to a probabilistic representation of solutions of a second-order PDE whose second order coefficients depend on the gradient of the solution. We then give some examples in dimension 1 and dimension 2 for which the assumptions are easy to check.
Classification : 60H10, 60H30
Mots clés : forward-backward stochastic differential equations, partial differential equations
@article{PS_2006__10__184_0, author = {Abraham, Romain and Riviere, Olivier}, title = {Forward-backward stochastic differential equations and {PDE} with gradient dependent second order coefficients}, journal = {ESAIM: Probability and Statistics}, pages = {184--205}, publisher = {EDP-Sciences}, volume = {10}, year = {2006}, doi = {10.1051/ps:2006005}, mrnumber = {2218408}, language = {en}, url = {http://www.numdam.org/articles/10.1051/ps:2006005/} }
TY - JOUR AU - Abraham, Romain AU - Riviere, Olivier TI - Forward-backward stochastic differential equations and PDE with gradient dependent second order coefficients JO - ESAIM: Probability and Statistics PY - 2006 DA - 2006/// SP - 184 EP - 205 VL - 10 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps:2006005/ UR - https://www.ams.org/mathscinet-getitem?mr=2218408 UR - https://doi.org/10.1051/ps:2006005 DO - 10.1051/ps:2006005 LA - en ID - PS_2006__10__184_0 ER -
Abraham, Romain; Riviere, Olivier. Forward-backward stochastic differential equations and PDE with gradient dependent second order coefficients. ESAIM: Probability and Statistics, Tome 10 (2006), pp. 184-205. doi : 10.1051/ps:2006005. http://www.numdam.org/articles/10.1051/ps:2006005/
[1] Backward forward stochastic differential equations. Ann. Appl. Probab. 3 (1993) 777-793. | Zbl 0780.60058
,[2] On the existence and uniqueness of solutions to fbsdes in a non-degenerate case. Stochastic Process. Appl. 99 (2002) 209-286. | Zbl 1058.60042
,[3] A forward-backward stochastic algorithm for quasi-linear PDEs. Ann. Appl. Probab. 16 (2006). | MR 2209339 | Zbl 1097.65011
and ,[4] Solving forward-backward stochastic differential equations explicitely - a four step scheme. Probab. Th. Rel. Fields 98 (1994) 339-359. | Zbl 0794.60056
, and ,[5] Forward-backward stochastic differential equations and their applications. Springer, Berlin. Lect. Notes Math. 1702 (1999). | MR 1704232 | Zbl 0927.60004
and ,[6] Backward stochastic differential equations and viscosity solutions of systems of semilinear parabolic and elliptic pdes of second order, in Stochastic Analysis and Relates Topics: The Geilo Workshop (1996) 79-127. | Zbl 0893.60036
,[7] Forward-backward stochastic differential equations and quasilinear parabolic pdes. Probab. Th. Rel. Fields 114 (1999) 123-150. | Zbl 0943.60057
and ,[8] Scale space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Machine Intell. 12 (1990) 629-639.
and ,Cité par Sources :