  By analogy with the theory of Backward Stochastic Differential Equations, we
define Backward Stochastic Difference Equations on spaces related to discrete
time, finite state processes. This paper considers these processes as
constructions in their own right, not as approximations to the continuous case.
We establish the existence and uniqueness of solutions under weaker assumptions
than are needed in the continuous time setting, and also establish a comparison
theorem for these solutions. The conditions of this theorem are shown to
approximate those required in the continuous time setting. We also explore the
relationship between the driver $F$ and the set of solutions; in particular, we
determine under what conditions the driver is uniquely determined by the
solution. Applications to the theory of nonlinear expectations are explored,
including a representation result.
