Backward Stochastic Difference Equations and Nearly Time-Consistent Nonlinear Expectations

Abstract

We consider backward stochastic difference equations (BSDEs) in discrete time with infinitely many states. This paper shows the existence and uniqueness of solutions to these equations in complete generality, and also derives a comparison theorem. Using these, time-consistent nonlinear evaluations and expectations are considered, and it is shown that every such evaluation or expectation corresponds to the solution of a BSDE without any requirements for continuity or boundedness. The implications of these results in a continuous time context are then considered, and possible applications are discussed.