Reflected Backward Stochastic Difference Equations with Finite State and their applications

TL;DR

This paper introduces reflected backward stochastic difference equations for finite state spaces, establishing their theoretical properties, connections to optimal stopping problems, and applications to American option pricing in various market conditions.

Contribution

It develops the theory of FS-RBSDEs, including existence, uniqueness, and comparison theorems, and links them to optimal stopping and American option pricing.

Findings

Established existence and uniqueness of FS-RBSDEs

Connected FS-RBSDEs to optimal stopping problems with multiple priors

Applied FS-RBSDEs to American option pricing in different market settings

Abstract

In this paper, we first establish the reflected backward stochastic difference 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 under Knightian uncertainty is a special case of our FS-RBSDEs. As a byproduct we develop the general theory of g-martingales in discrete 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.