Explicit Runge-Kutta schemes for Backward Stochastic Differential Equations

TL;DR

This paper introduces explicit Runge-Kutta schemes for backward stochastic differential equations, extending classical Butcher theory to analyze their order conditions systematically and supporting the schemes with numerical validation.

Contribution

It develops a new class of explicit Runge-Kutta schemes for BSDEs and extends Butcher theory to derive order conditions systematically.

Findings

Schemes admit a concise formulation similar to ODE counterparts

Order conditions derived symbolically via extended Butcher theory

Numerical experiments confirm theoretical accuracy

Abstract

The Butcher theory provides a powerful tool for analyzing order conditions of Runge-Kutta schemes for ordinary differential equations (ODEs); however, such a theory has not yet been well established for backward stochastic differential equations (BSDEs) -- motivating the current work to address this gap. Specifically, we propose a new class of explicit Runge-Kutta schemes for BSDEs. These schemes admit a concise formulation that closely mirrors their ODE counterparts. Building on this formulation, we extend the Butcher theory to the proposed schemes, thereby enabling a symbolic derivation of Taylor expansions for the local truncation errors, and yielding the order conditions. Our approach preserves the elegance and generality of the original Butcher theory: it avoids stage-by-stage error expansions and provides a systematic, stage-inductive analysis, applicable to schemes with any number of stages and any target order. Numerical experiments support the theoretical results.