Forward-backward stochastic differential equations and PDE with gradient dependent second order coefficients
ESAIM: Probability and Statistics, Tome 10 (2006), pp. 184-205.

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.

DOI : https://doi.org/10.1051/ps:2006005
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] F. Antonelli, Backward forward stochastic differential equations. Ann. Appl. Probab. 3 (1993) 777-793. | Zbl 0780.60058

[2] F. Delarue, 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] F. Delarue and S. Menozzi, A forward-backward stochastic algorithm for quasi-linear PDEs. Ann. Appl. Probab. 16 (2006). | MR 2209339 | Zbl 1097.65011

[4] J. Ma, P. Protter and J. Yong, Solving forward-backward stochastic differential equations explicitely - a four step scheme. Probab. Th. Rel. Fields 98 (1994) 339-359. | Zbl 0794.60056

[5] J. Ma and J. Yong, Forward-backward stochastic differential equations and their applications. Springer, Berlin. Lect. Notes Math. 1702 (1999). | MR 1704232 | Zbl 0927.60004

[6] E. Pardoux, 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] E. Pardoux and S. Tang, Forward-backward stochastic differential equations and quasilinear parabolic pdes. Probab. Th. Rel. Fields 114 (1999) 123-150. | Zbl 0943.60057

[8] P. Perona and J. Malik, Scale space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Machine Intell. 12 (1990) 629-639.

Cité par Sources :