Infinite Horizon Fully Coupled Nonlinear Forward-Backward Stochastic Difference Equations and Their Application to LQ Optimal Control Problems

TL;DR

This paper establishes the unique solvability of infinite horizon fully coupled nonlinear forward-backward stochastic difference equations under new domination-monotonicity conditions, with applications to linear quadratic optimal control problems.

Contribution

It introduces a novel approach using domination-monotonicity conditions to prove solvability of coupled FBSΔEs in infinite horizon discrete-time systems, extending previous methods.

Findings

Proved unique solvability of coupled FBSΔEs under new conditions.

Derived solution estimates for the equations.

Applied results to characterize optimal controls in LQ problems.

Abstract

This paper focuses on the study of infinite horizon fully coupled nonlinear forward-backward stochastic difference equations (FBS$\bigtriangleup$Es). Firstly, we establish a pair of priori estimates for the solutions to forward stochastic difference equations (S$\bigtriangleup$Es) and backward stochastic difference equations (BS$\bigtriangleup$Es), respectively. Then, to achieve broader applicability, we utilize a set of domination-monotonicity conditions that are more lenient than standard assumptions. Using these conditions, we apply continuation methods to prove the unique solvability of infinite horizon fully coupled FBS$\bigtriangleup$Es and derive a set of solution estimates. Furthermore, our results have considerable implications for a variety of related linear quadratic (LQ) problems, especially when the stochastic Hamiltonian system is consistent with FBS$\bigtriangleup$Es satisfying the introduced domination-monotonicity conditions. Thus, by solving the associated stochastic Hamiltonian system, we explicitly characterize the unique optimal control. This is the first work establishing solvability of fully coupled nonlinear FBS$\bigtriangleup$Es under domination-monotonicity conditions in infinite horizon discrete-time setting.