Geodesic
Forward-backward stochastic differential equations and PDE with gradient dependent second order coefficients
ESAIM: Probability and Statistics, Tome 10 (2006), pp. 184-205

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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.

DOI : 10.1051/ps:2006005
Classification : 60H10, 60H30
Keywords: forward-backward stochastic differential equations, partial differential equations 
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     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://geodesic.mathdoc.fr/articles/10.1051/ps:2006005/}
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Abraham, Romain; Riviere, Olivier. Forward-backward stochastic differential equations and PDE with gradient dependent second order coefficients. ESAIM: Probability and Statistics, Tome 10 (2006), pp. 184-205. doi: 10.1051/ps:2006005

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