  In this work, we concern with the high order numerical methods for coupled
forward-backward stochastic differential equations (FBSDEs). Based on the
FBSDEs theory, we derive two reference ordinary differential equations (ODEs)
from the backward SDE, which contain the conditional expectations and their
derivatives. Then, our high order multi-step schemes are obtained by carefully
approximating the derivatives and the conditional expectations in the reference
ODEs. Motivated by the local property of the generator of diffusion processes,
the Euler method is used to solve the forward SDE, however, it is noticed that
the numerical solution of the backward SDE is still of high order accuracy.
Such results are obviously promising: on one hand, the use of Euler method (for
the forward SDE) can dramatically simplifies the entire computational scheme,
and on the other hand, one might be only interested in the solution of the
backward SDE in many real applications such as option pricing. Several
numerical experiments are carried out to demonstrate the effectiveness of the
numerical method.
