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arXiv · 2410.01749

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

Zhipeng Niu·Qingxin Meng·Xun Li·Maoning Tang
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Abstract

This paper explores a class of fully coupled nonlinear forward-backward stochastic difference equations (FBS$Δ$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$Δ$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$Δ$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.

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Zhipeng Niu, Qingxin Meng, Xun Li, Maoning Tang. 2025-07-31. Fully Coupled Nonlinear FBS$Δ$Es: Solvability and LQ Control Insights. https://arxiv.org/abs/2410.01749

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