Forward-backward stochastic differential equations and PDE with gradient dependent second order coefficients
Laboratoire MAPMO, Université d'Orléans, B.P. 6759,
45067 Orléans Cedex 2,
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
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  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