Solvability of finite state forward-backward stochastic difference equations

TL;DR

This paper investigates the conditions under which fully coupled forward-backward stochastic difference equations (FBS{ extDelta}Es) are solvable in discrete time, finite state spaces, providing criteria for both linear and nonlinear cases.

Contribution

It establishes necessary and sufficient conditions for linear FBS{ extDelta}Es and proves existence and uniqueness results for nonlinear FBS{ extDelta}Es under monotone conditions.

Findings

Necessary and sufficient conditions for linear FBS{ extDelta}Es solvability

Existence and uniqueness theorems for nonlinear FBS{ extDelta}Es

Results applicable to discrete time, finite state processes

Abstract

In this paper, we consider the solvability problems for the fully coupled forward-backward stochastic difference equations (FBS{\Delta}Es) on spaces related to discrete time, finite state processes. On one hand, we provide the necessary and sufficient condition for the solvability of the linear FBS{\Delta}Es. On the other hand, under the assumption that the coefficients satisfy the monotone condition, we investigate the existence and uniqueness theorems for the general nonlinear FBS{\Delta}Es.