  An approach for the description of stochastic systems is derived. Some of the
variables in the system are studied forward in time, others backward in time.
The approach is based on a perturbation expansion in the strength of the
coupling between forward and backward variables, and is well suited for
situations in which initial and final conditions are imposed on different
components of the system and the coupling between those components is weak. The
form of the stochastic equations in our approach is determined by requiring
that they generate the same statistics obtained in a forward description of the
dynamics. Numerical tests are carried out on a few simple two-degree-of-freedom
systems. The merit and the difficulties of the approach are discussed and
compared to more traditional strategies based on transition path sampling and
simple shooting algorithms.
