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.

Keywords: 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 SP - 184 EP - 205 VL - 10 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps:2006005/ DO - 10.1051/ps:2006005 LA - en ID - PS_2006__10__184_0 ER -

%0 Journal Article %A Abraham, Romain %A Riviere, Olivier %T Forward-backward stochastic differential equations and PDE with gradient dependent second order coefficients %J ESAIM: Probability and Statistics %D 2006 %P 184-205 %V 10 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps:2006005/ %R 10.1051/ps:2006005 %G en %F PS_2006__10__184_0

Abraham, Romain; Riviere, Olivier. Forward-backward stochastic differential equations and PDE with gradient dependent second order coefficients. ESAIM: Probability and Statistics, Volume 10 (2006), pp. 184-205. doi : 10.1051/ps:2006005. http://www.numdam.org/articles/10.1051/ps:2006005/

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