A forward differential deep learning-based algorithm for solving high-dimensional nonlinear backward stochastic differential equations

Paper: arXiv 2408.05620

Abstract

In this work, we present a novel forward differential deep learning-based algorithm for solving high-dimensional nonlinear backward stochastic differential equations (BSDEs). Motivated by the fact that differential deep learning can efficiently approximate the labels and their derivatives with respect to inputs, we transform the BSDE problem into a differential deep learning problem. This is done by leveraging Malliavin calculus, resulting in a system of BSDEs. The unknown solution of the BSDE system is a triple of processes $(Y, Z, Γ)$, representing the solution, its gradient, and the Hessian matrix. The main idea of our algorithm is to discretize the integrals using the Euler-Maruyama method and approximate the unknown discrete solution triple using three deep neural networks. The parameters of these networks are then optimized by globally minimizing a differential learning loss function, which is novelty defined as a weighted sum of the dynamics of the discretized system of BSDEs. Through various high-dimensional examples, we demonstrate that our proposed scheme is more efficient in terms of accuracy and computation time compared to other contemporary forward deep learning-based methodologies.

Complexity vs Empirical Score

• Math Complexity: 9.5/10
• Empirical Rigor: 7.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper is highly theoretical, employing advanced concepts like Malliavin calculus, high-dimensional BSDE systems, and detailed network architecture derivations, warranting a high math score. Empirical rigor is strong, with comparisons to contemporary methods and specific use cases like option pricing, but lacks the code or raw datasets that would push it to the highest tier.

Research Flowchart

  flowchart TD
  Start["Research Goal: Solve High-Dimensional Nonlinear Backward Stochastic Differential Equations BSDEs"] --> Method["Methodology: Forward Differential Deep Learning Algorithm"]
  
  Method --> Inputs["Data/Inputs: High-Dimensional BSDEs, Malliavin Calculus Framework"]
  Method --> Proc["Computational Process: Discretize using Euler-Maruyama, Approximate (Y, Z, Γ) via 3 DNNs, Optimize via Differential Loss"]
  
  Inputs --> Proc
  
  Proc --> Findings["Key Findings: Increased Accuracy & Computation Time Efficiency vs. Contemporary Methods"]
  
  Findings --> End["Outcome: Effective Solver for High-Dimensional Nonlinear BSDEs"]

Cited References (39)

1. Adapted solution of a backward stochastic differential equation (1990) — link
2. Backward Stochastic Differential Equations in Finance (1997) — link
3. Uncertainty quantification for deep learning-based schemes for solving high-dimensional backward stochastic differential equations (2023) — link
4. A learning scheme by sparse grids and Picard approximations for semilinear parabolic PDEs (2023) — link
5. Deep Learning-Based Numerical Methods for High-Dimensional Parabolic Partial Differential Equations and Backward Stochastic Differential Equations (2017) — link
6. Solving high-dimensional partial differential equations using deep learning (2018) — link
7. Forward–backward stochastic neural networks: deep learning of high-dimensional partial differential equations (2024) — link
8. Asymptotic Expansion as Prior Knowledge in Deep Learning Method for High dimensional BSDEs (2019) — link

0 of 39 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.