  In this paper, an analytic approximation method for highly nonlinear
equations, namely the homotopy analysis method (HAM), is employed to solve some
backward stochastic differential equations (BSDEs) and forward-backward
stochastic differential equations (FBSDEs), including one with high
dimensionality (up to 12 dimensions). By means of the HAM, convergent series
solutions can be quickly obtained with high accuracy for a FBSDE in a 6
dimensional case, within less than $1\%$ CPU time used by a currently reported
numerical method for the same case [34]. Especially, as dimensionality
enlarges, the increase of computational complexity for the HAM is not as
dramatic as this numerical method. All of these demonstrate the validity and
high efficiency of the HAM for the backward/forward-backward stochastic
differential equations in science, engineering and finance.
