[Paper Review] Three algorithms for solving high-dimensional fully-coupled FBSDEs through deep learning

Shaolin Ji, Shigē Péng+2 more|arXiv (Cornell University)|Jul 11, 2019

Model Reduction and Neural Networks29 references4 citations

TL;DR

This paper proposes three deep learning-based algorithms for solving high-dimensional fully-coupled forward-backward stochastic differential equations (FBSDEs), leveraging neural networks to approximate the control variable Z with different feedback structures. The methods overcome the curse of dimensionality, with Algorithm 3 showing superior accuracy and stability, achieving relative errors below 0.07% in 100-dimensional tests.

ABSTRACT

Recently, the deep learning method has been used for solving forward-backward stochastic differential equations (FBSDEs) and parabolic partial differential equations (PDEs). It has good accuracy and performance for high-dimensional problems. In this paper, we mainly solve fully coupled FBSDEs through deep learning and provide three algorithms. Several numerical results show remarkable performance especially for high-dimensional cases.

Motivation & Objective

• Address the challenge of solving high-dimensional fully-coupled FBSDEs, which suffer from the curse of dimensionality in classical numerical methods.
• Develop deep learning-based numerical schemes that can efficiently approximate solutions to FBSDEs in high dimensions.
• Design three distinct neural network architectures with different feedback mechanisms for the control variable Z to improve accuracy and convergence.
• Demonstrate the effectiveness of these algorithms on benchmark FBSDEs with up to 100 dimensions, showing superior performance over traditional methods.
• Provide a systematic comparison of the three algorithms in terms of convergence speed, accuracy, and computational cost.

Proposed method

• Formulate the FBSDE solution as an optimal control problem, where the Z process is treated as a control input to be learned by a neural network.
• Design Algorithm 1 with Z dependent on forward state X and backward state Y, extending prior deep learning methods to fully-coupled cases.
• Introduce Algorithm 2 by treating Y as a control variable u, with both u and Z dependent on X, and add a penalty term to enforce consistency between u and Y.
• Propose Algorithm 3 based on Picard iteration, where Z at step k+1 depends on X^{k+1}, Y^k, and Z^k, enabling path-dependent feedback.
• Train the neural networks via stochastic gradient descent by minimizing a cost functional that includes terminal condition and dynamic constraints.
• Use Monte Carlo sampling to approximate the Itô integrals in the FBSDE and compute the loss function during training.

Experimental results

Research questions

• RQ1 Can deep learning effectively solve high-dimensional fully-coupled FBSDEs that are intractable with classical numerical methods?
• RQ2 How do different feedback structures for the control variable Z impact convergence speed, accuracy, and stability?
• RQ3 What is the relative performance of the three proposed algorithms in terms of error, variance, and computational cost across varying dimensions?
• RQ4 Does the path-dependent feedback in Algorithm 3 lead to faster convergence and better accuracy compared to state-feedback approaches?
• RQ5 Can the methods achieve high accuracy (e.g., relative error < 0.1%) in 100-dimensional FBSDEs with complex nonlinear generators?

Key findings

• Algorithm 3 achieves the highest accuracy and stability, with a relative error of only 4.24×10⁻⁴ in a 10-dimensional test case.
• In the 100-dimensional test case, Algorithm 3 achieved a relative error of 0.061% after 4000 training steps, closely matching the explicit solution.
• Algorithm 3 required the fewest iteration steps (1087.8 on average) to converge, though each step took longer than in Algorithms 1 and 2.
• Algorithm 2 showed the lowest variance in numerical results (1.262×10⁻⁹ in Example 4), indicating high stability, but failed to converge stably in Example 3.
• Algorithms 1 and 2 were computationally faster per step but required more iterations to converge, especially in high-dimensional settings.
• All three algorithms achieved relative errors below 1% across all test cases, demonstrating robustness and scalability to high-dimensional problems.