Fully Coupled Nonlinear FBSΔEs: Solvability and LQ Control Insights

2024/10/02 by Niu, Zhipeng, Meng, Qingxin, Li, Xun +1
#FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2410.01749

Abstract

This paper explores a class of fully coupled nonlinear forward-backward stochastic difference equations (FBSΔEs). 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 FBSΔEs 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 FBSΔEs 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.

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