  In this work, we have presented a simple analytical approximation scheme for
generic non-linear FBSDEs. By treating the interested system as the linear
decoupled FBSDE perturbed with non-linear generator and feedback terms, we have
shown that it is possible to carry out a recursive approximation to an
arbitrarily higher order, where the required calculations in each order are
equivalent to those for standard European contingent claims. We have also
applied the perturbative method to the PDE framework following the so-called
Four Step Scheme. The method is found to render the original non-linear PDE
into a series of standard parabolic linear PDEs. Due to the equivalence of the
two approaches, it is also possible to derive approximate analytic solution for
the non-linear PDE by applying the asymptotic expansion to the corresponding
probabilistic model. Two simple examples are provided to demonstrate how the
perturbation works and show its accuracy relative to known numerical
techniques. The method presented in this paper may be useful for various
important problems which have eluded analytical treatment so far.
