This paper focuses on the stability analysis of a general class of linear multistep methods for decoupled forward–backward stochastic differential equations (FBSDEs). The general linear multistep methods we consider contain many well-known linear multistep methods from the ordinary differential equation framework, such as Adams, Nyström, Milne--Simpson and backward differentiation formula methods. Under the classical root condition, we prove that general linear multistep methods are mean-square (zero) stable for decoupled FBSDEs with generator function related to both y and z. Based on the stability result, we further establish a fundamental convergence theorem.

中文翻译：

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马尔可夫后向随机微分方程一般多步法的稳定性分析

本文重点研究解耦正反向随机微分方程 (FBSDE) 的一般线性多步方法的稳定性分析。我们考虑的一般线性多步方法包含许多来自常微分方程框架的众所周知的线性多步方法，例如 Adams、Nyström、Milne-Simpson 和后向微分公式方法。在经典根条件下，我们证明了一般线性多步方法对于具有与 y 和 z 相关的生成函数的解耦 FBSDE 是均方（零）稳定的。基于稳定性结果，我们进一步建立了一个基本收敛定理。