  An approach for the description of stochastic systems is derived, in which
some of the variables in the system are studied forward in time and 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 in different parts
of the system are imposed. The stochastic equations in the new approach are
determined by requiring that the same statistics that would be obtained in a
forward description of the dynamics is reproduced. Numerical tests are carried
out on a few simple two-degree-of-freedom systems. The merit and the
difficulties of the approach are discussed compared to more traditional
strategies based on transition path sampling and simple shooting algorithms.
