Stability analysis of general multistep methods for Markovian backward stochastic differential equations

Abstract

Abstract 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.