# Forward-backward stochastic differential equations and PDE with gradient dependent second order coefficients

Romain Abraham; Olivier Riviere

ESAIM: Probability and Statistics (2006)

- Volume: 10, page 184-205
- ISSN: 1292-8100

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topAbraham, Romain, and Riviere, Olivier. "Forward-backward stochastic differential equations and PDE with gradient dependent second order coefficients." ESAIM: Probability and Statistics 10 (2006): 184-205. <http://eudml.org/doc/249748>.

@article{Abraham2006,

abstract = {
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.
},

author = {Abraham, Romain, Riviere, Olivier},

journal = {ESAIM: Probability and Statistics},

keywords = {Forward-backward stochastic differential equations; partial differential equations.; forward-backward stochastic differential equations; partial differential equations},

language = {eng},

month = {3},

pages = {184-205},

publisher = {EDP Sciences},

title = {Forward-backward stochastic differential equations and PDE with gradient dependent second order coefficients},

url = {http://eudml.org/doc/249748},

volume = {10},

year = {2006},

}

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

DA - 2006/3//

PB - EDP Sciences

VL - 10

SP - 184

EP - 205

AB -
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.

LA - eng

KW - Forward-backward stochastic differential equations; partial differential equations.; forward-backward stochastic differential equations; partial differential equations

UR - http://eudml.org/doc/249748

ER -

## References

top- F. Antonelli, Backward forward stochastic differential equations. Ann. Appl. Probab.3 (1993) 777–793. Zbl0780.60058
- F. Delarue, On the existence and uniqueness of solutions to fbsdes in a non-degenerate case. Stochastic Process. Appl.99 (2002) 209–286. Zbl1058.60042
- F. Delarue and S. Menozzi, A forward-backward stochastic algorithm for quasi-linear PDEs. Ann. Appl. Probab.16 (2006). Zbl1097.65011
- J. Ma, P. Protter and J. Yong, Solving forward-backward stochastic differential equations explicitely – a four step scheme. Probab. Th. Rel. Fields98 (1994) 339–359. Zbl0794.60056
- J. Ma and J. Yong, Forward-backward stochastic differential equations and their applications. Springer, Berlin. Lect. Notes Math.1702 (1999). Zbl0927.60004
- 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. Zbl0893.60036
- E. Pardoux and S. Tang, Forward-backward stochastic differential equations and quasilinear parabolic pdes. Probab. Th. Rel. Fields114 (1999) 123–150. Zbl0943.60057
- P. Perona and J. Malik, Scale space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Machine Intell.12 (1990) 629–639.

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