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

Romain Abraham (1) , Olivier Rivière (2)

1 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans
2 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5

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.

Domaines: Probabilités [math.PR]

Romain Abraham, Olivier Rivière. Forward-backward stochastic differential equations and PDE with gradient dependent second order coefficients. ESAIM: Probability and Statistics, 2006, 10, pp.184-205. ⟨hal-00008569⟩