[Paper Review] Numerical methods for backward stochastic differential equations: A survey

Jared Chessari, Reiichiro Kawai+2 more|arXiv (Cornell University)|Jan 22, 2021

Stochastic processes and financial applications92 references4 citations

TL;DR

This survey provides a comprehensive, systematic comparison of numerical methods for backward stochastic differential equations (BSDEs), categorizing 333 references into backward, forward, and deep learning-based approaches. It evaluates key assumptions, convergence properties, and practical trade-offs, offering a unified framework for selecting and advancing methods, especially for high-dimensional and nonlinear problems.

ABSTRACT

Backward Stochastic Differential Equations (BSDEs) have been widely employed in various areas of social and natural sciences, such as the pricing and hedging of financial derivatives, stochastic optimal control problems, optimal stopping problems and gene expression. Most BSDEs cannot be solved analytically and thus numerical methods must be applied to approximate their solutions. There have been a variety of numerical methods proposed over the past few decades as well as many more currently being developed. For the most part, they exist in a complex and scattered manner with each requiring a variety of assumptions and conditions. The aim of the present work is thus to systematically survey various numerical methods for BSDEs, and in particular, compare and categorize them, for further developments and improvements. To achieve this goal, we focus primarily on the core features of each method based on an extensive collection of 333 references: the main assumptions, the numerical algorithm itself, key convergence properties and advantages and disadvantages, to provide an up-to-date coverage of numerical methods for BSDEs, with insightful summaries of each and a useful comparison and categorization.

Motivation & Objective

• To address the fragmented and scattered state of numerical methods for BSDEs by providing a unified, systematic survey.
• To compare and categorize existing numerical methods based on core features such as assumptions, algorithms, convergence, and practical advantages/disadvantages.
• To support future methodological development by identifying gaps, especially for nonstandard BSDE types like those with jumps, reflection, or non-global Lipschitz conditions.
• To highlight the transformative potential of deep learning-based methods for high-dimensional and nonlinear BSDEs.
• To serve as a foundational reference for researchers selecting or designing numerical schemes for BSDEs in finance, control, and PDEs.

Proposed method

• Categorizes numerical methods into three main classes: backward methods (e.g., backward Euler, higher-order schemes), forward methods (e.g., Picard iteration, multilevel Picard), and deep learning-based methods.
• Analyzes computation of conditional expectations—critical for backward methods—using least-squares regression, Malliavin calculus, quantization, tree-based, and cubature methods.
• Evaluates forward methods via Picard iteration, branching diffusion systems, asymptotic expansion, and multilevel Picard approximations.
• Examines deep learning-based approaches including Deep BSDE, deep backward dynamic programming, deep splitting, and physics-informed neural networks.
• Uses a comparative framework to assess methods across convergence rates, computational cost, dimensionality scalability, and robustness to non-smooth drivers.
• Incorporates theoretical and numerical insights from 333 references to evaluate methodological strengths and limitations in diverse settings.

Experimental results

Research questions

• RQ1 What are the key assumptions, convergence properties, and practical trade-offs of backward Euler and higher-order discretization methods for BSDEs?
• RQ2 How do different techniques for computing conditional expectations (e.g., least-squares regression, Malliavin calculus) compare in accuracy and efficiency?
• RQ3 In what ways do forward methods such as Picard iteration and multilevel Picard approximations improve scalability for high-dimensional BSDEs?
• RQ4 How do deep learning-based methods overcome limitations of classical numerical schemes in handling high-dimensional and nonlinear BSDEs?
• RQ5 What are the current theoretical and numerical challenges in solving BSDEs with nonstandard features such as jumps, reflection, or quadratic drivers?

Key findings

• Deep learning-based methods, such as Deep BSDE and deep splitting, demonstrate strong potential for solving high-dimensional BSDEs and semilinear PDEs, where classical methods fail due to the curse of dimensionality.
• Least-squares regression remains a widely used and effective method for computing conditional expectations, though its accuracy depends on basis function selection and may degrade in high dimensions.
• Malliavin calculus-based methods offer a theoretically robust alternative for conditional expectation computation, particularly when the driver is non-Markovian or irregular.
• Multilevel Picard approximations achieve exponential convergence rates under mild conditions, making them highly efficient for certain classes of semilinear PDEs and BSDEs.
• The survey identifies significant research gaps in numerical methods for ergodic, delayed, and regime-switching BSDEs, indicating unmet needs in theoretical and applied research.
• Despite advances, theoretical convergence analysis for many deep learning-based methods remains incomplete, highlighting a critical area for future work.