A general theory of Finite State Backward Stochastic Difference Equations

TL;DR

This paper develops a comprehensive theory for Backward Stochastic Difference Equations in discrete time and finite state spaces, establishing foundational results like existence, uniqueness, and comparison theorems, with applications to nonlinear expectations.

Contribution

It introduces a general framework for BSDEs in discrete finite state spaces, proving key properties without relying on continuous approximations, and explores their relation to nonlinear expectations.

Findings

Proved existence and uniqueness of solutions under weaker assumptions.

Established a comparison theorem for solutions.

Linked the driver function to the set of solutions and nonlinear expectations.

Abstract

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.