Stochastic maximum principle for fully coupled nonlinear FBSΔEs under generalized monotonicity and LQ control applications

Abstract

This paper investigates the optimal control problem for a class of nonlinear fully coupled forward–backward stochastic difference equations (FBSΔEs). Under the convexity assumption of the control domain, we establish a variational formula for the cost functional involving the Hamiltonian system and adjoint equations, deriving both necessary and sufficient optimality conditions via the Pontryagin maximum principle. Innovatively, we employ a generalized monotonicity framework to ensure the existence and uniqueness of solutions for nonlinear systems and directly derive variational inequalities through the convexity properties of the Hamiltonian function, simplifying the analysis of fully coupled systems. As an application, we formulate a linear-quadratic (LQ) optimal control problem inspired by energy storage scheduling (a real-world example) to demonstrate the effectiveness of our theoretical results. The study reveals that discrete-time FBSΔEs models offer significant computational advantages for practical systems with future-dependent constraints, such as power dispatch and financial decision-making, providing a new theoretical foundation for high-dimensional optimal control problems.