Discrete Time And Finite State Reflected Backward Stochastic Difference Equations And Their Applications

The theory of backward stochastic differential equations(BSDEs) was first introduced by Pardoux and Peng [16]. Over the past twenty years. BS-DEs are widely used in mathematical finance, stochastic control and other fields. By analogy with the theory of BSDEs, Cohen and Elliott [8] consid-ered the backward stochastic difference equations related to discrete time and finite state processes as entities in their own right, not as approxima-tions to the continuous case.The general theory of reflected backward stochastic differential equa-tions (RBSDEs for short) was studied by El Karoui et al. [9]. This im-portant theory could be applied to the optimal stopping problem and some problems in finance markets [10]. So it is interesting to explore the reflected BSDEs in the framework of [8], as well as some applications.In this paper, we first establish the reflected backward stochastic differ-ence equations with finite state (FS-RBSDEs for short). Then we explore the Existence and Uniqueness Theorem as well as the Comparison Theorem by "one step" method. The connections between FS-RBSDEs and optimal stopping time problems are investigated and we also show that the optimal stopping problems with multiple priors is a special case of our FS-RBSDEs, as well as the connections between the FS-BSDEs and the multiple prior martingale under Knightian uncertainty.As a byproduct we develop the general theory of g-martingales in dis-crete time with finite state including Doob-Mayer Decomposition Theorem and Optional Sampling Theorem. Finally, we consider the pricing models of American Option in both complete and incomplete markets.