| Issue |
ESAIM: PS
Volume 10, September 2006
|
|
|---|---|---|
| Page(s) | 184 - 205 | |
| DOI | https://doi.org/10.1051/ps:2006005 | |
| Published online | 09 March 2006 | |
Forward-backward stochastic differential equations and PDE with gradient dependent second order coefficients
1
Laboratoire MAPMO, Université d'Orléans, B.P. 6759,
45067 Orléans Cedex 2,
France;
Romain.Abraham@univ-orleans.fr
2
Laboratoire MAP5, UFR de Mathématiques et d'Informatique, Université René
Descartes, 45 rue des Saints Pères, 75270 Paris Cedex 06, France;
Olivier.Riviere@math-info.univ-paris5.fr
Received:
17
June
2004
Revised:
17
June
2005
Revised:
12
September
2005
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.
Mathematics Subject Classification: 60H10 / 60H30
Key words: Forward-backward stochastic differential equations / partial differential equations.
© EDP Sciences, SMAI, 2006
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