Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fully Coupled Nonlinear FBS$Δ$Es: Maximum principle and LQ Control Insights

Zhipeng Niu, Jun Moon, Qingxin Meng

TL;DR

The paper develops a stochastic maximum principle for nonlinear fully coupled forward-backward stochastic difference equations (FBSΔEs) in discrete time, under convex control domains. It derives a variational framework centered on a Hamiltonian and adjoint BSΔE, establishing both necessary and sufficient optimality conditions. An energy-storage motivated discrete-time linear-quadratic example demonstrates the practicality of the approach and yields explicit control laws. The work unifies solvability results for nonlinear FBΔEs with LQ control, offering a foundation for extensions to mean-field settings, data-driven methods, and scalable numerical schemes.

Abstract

This paper investigates the optimal control problem for a class of nonlinear fully coupled forward-backward stochastic difference equations (FBS$Δ$Es). Under the convexity assumption of the control domain, we establish a variational formula for the cost functional involving the Hamiltonian and adjoint system. Both necessary and sufficient conditions for optimal control are derived using the Pontryagin maximum principle. As an application, we present a linear quadratic optimal control problem to illustrate our theoretical results.

Fully Coupled Nonlinear FBS$Δ$Es: Maximum principle and LQ Control Insights

TL;DR

The paper develops a stochastic maximum principle for nonlinear fully coupled forward-backward stochastic difference equations (FBSΔEs) in discrete time, under convex control domains. It derives a variational framework centered on a Hamiltonian and adjoint BSΔE, establishing both necessary and sufficient optimality conditions. An energy-storage motivated discrete-time linear-quadratic example demonstrates the practicality of the approach and yields explicit control laws. The work unifies solvability results for nonlinear FBΔEs with LQ control, offering a foundation for extensions to mean-field settings, data-driven methods, and scalable numerical schemes.

Abstract

This paper investigates the optimal control problem for a class of nonlinear fully coupled forward-backward stochastic difference equations (FBSEs). Under the convexity assumption of the control domain, we establish a variational formula for the cost functional involving the Hamiltonian and adjoint system. Both necessary and sufficient conditions for optimal control are derived using the Pontryagin maximum principle. As an application, we present a linear quadratic optimal control problem to illustrate our theoretical results.

Paper Structure

This paper contains 7 sections, 8 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. Stochastic maximum principle for the coupled FBS$\bigtriangleup$E system
  4. Application in linear quadratic problem
  5. Problem Background and Modeling Motivation
  6. Reformulation as a Linear-Quadratic Problem
  7. Conclusion

Key Result

Lemma 2.2

Let $(\Lambda, \Phi, \Gamma)$ be a set of coefficients satisfying Assumptions ass:2.3 and ass:2.2. Then, for any $u(\cdot) \in \mathbb{U}$, the FBS$\Delta$E eq:2.1 admits a unique solution $(x(\cdot), y(\cdot)) \in N^{2}(\overline{\mathbb{T}}; \mathbb{R}^{2n})$. Moreover, the following estimate hold where and $K$ is a positive constant depending only on $\mathbb{T}$, the Lipschitz constants, $\mu

Theorems & Definitions (15)

  • Lemma 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • Theorem 3.5
  • ...and 5 more