Fully Coupled Nonlinear FBS$\Delta$Es: Solvability and LQ Control Insights

TL;DR

This paper establishes the unique solvability of fully coupled nonlinear forward-backward stochastic difference equations using domination-monotonicity conditions and applies these results to derive explicit solutions for related linear quadratic control problems.

Contribution

Introduces a relaxed domination-monotonicity framework for discrete systems and demonstrates its effectiveness in solving fully coupled nonlinear FBSΔEs and related LQ control problems.

Findings

Proves unique solvability of fully coupled nonlinear FBSΔEs.

Derives explicit optimal controls for related LQ problems.

Provides solution estimates under new domination-monotonicity conditions.

Abstract

This paper explores a class of fully coupled nonlinear forward-backward stochastic difference equations (FBS$\Delta$Es). Building on insights from linear quadratic optimal control problems, we introduce a more relaxed framework of domination-monotonicity conditions specifically designed for discrete systems. Utilizing these conditions, we apply the method of continuation to demonstrate the unique solvability of the fully coupled FBS$\Delta$Es and derive a set of solution estimates. Moreover, our results have considerable implications for various related linear quadratic (LQ) problems, particularly where stochastic Hamiltonian systems are aligned with the FBS$\Delta$Es meeting these introduced domination-monotonicity conditions. As a result, solving the associated stochastic Hamiltonian systems allows us to derive explicit expressions for the unique optimal controls.