Discrete-Time BSDEs with Random Terminal Horizon

This article studies the theory of discrete-time backward stochastic differential
equations (also called BSDEs) with a random terminal time, which is not a stopping
time. We follow Cohen and Elliott [2] and consider a reference filtration generated
by a general discrete-time finite-state process. The martingale representation
theorem for essentially bounded martingales under progressively enlarged filtration
is established. Then we prove the existence and uniqueness theorem of BSDEs under
enlarged filtration using some weak assumptions of the driver. We also present
conditions for a comparison theorem. Applications to nonlinear expectations and
optimal design of dynamic default risk are explored.