A Unified Probabilistic Discretization Scheme for FBSDEs: Stability, Consistency, and Convergence Analysis

In this work, we propose a general discretization framework for numerical solution of forward backward stochastic differential equations (FBSDEs). The framework covers several existing temporal discretization probabilistic schemes in the literature. The consistency, stability, and convergence analysis for the proposed scheme are presented. In particular, we prove a stochastic mean square version of Lax equivalence theorem-showing that a consistent discretization scheme for FBSDEs is convergent if and only if it is stable. Applications of the analysis results to existing numerical schemes are also discussed.