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This paper extends the results of Ma, Wu, Zhang, Zhang [11] to the context ofpath-dependent multidimensional forward-backward stochastic differentialequations (FBSDE). By path-dependent we mean that the coefficients of theforward-backward SDE at time t can depend on the whole path of the forwardprocess up to time t. Such a situation appears when solving path-dependentstochastic control problems by means of variational calculus. At the heart ofour analysis is the construction of a decoupling random field on the pathspace. We first prove the existence and the uniqueness of decoupling field onsmall time interval. Then by introducing the characteristic BSDE, we show thata global decoupling field can be constructed by patching local solutionstogether as long as the solution of the characteristic BSDE remains bounded.Finally, we provide a stability result for path-dependent forward-backwardSDEs.

On path-dependent multidimensional forward-backward SDEs