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

Paper: arXiv 2404.08456

Abstract

In this work, we propose a novel backward differential deep learning-based algorithm for solving high-dimensional nonlinear backward stochastic differential equations (BSDEs), where the deep neural network (DNN) models are trained not only on the inputs and labels but also the differentials of the corresponding labels. This is motivated by the fact that differential deep learning can provide an efficient approximation of the labels and their derivatives with respect to inputs. The BSDEs are reformulated as differential deep learning problems by using Malliavin calculus. The Malliavin derivatives of solution to a BSDE satisfy themselves another BSDE, resulting thus in a system of BSDEs. Such formulation requires the estimation of the solution, its gradient, and the Hessian matrix, represented by the triple of processes $\left(Y, Z, Γ\right).$ All the integrals within this system are discretized by using the Euler-Maruyama method. Subsequently, DNNs are employed to approximate the triple of these unknown processes. The DNN parameters are backwardly optimized at each time step by minimizing a differential learning type loss function, which is defined as a weighted sum of the dynamics of the discretized BSDE system, with the first term providing the dynamics of the process $Y$ and the other the process $Z$. An error analysis is carried out to show the convergence of the proposed algorithm. Various numerical experiments up to $50$ dimensions are provided to demonstrate the high efficiency. Both theoretically and numerically, it is demonstrated that our proposed scheme is more efficient compared to other contemporary deep learning-based methodologies, especially in the computation of the process $Γ$.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper is extremely heavy on advanced mathematics, including high-dimensional BSDEs, Malliavin calculus, and formal error analysis, while the empirical validation is limited to controlled numerical experiments in up to 50 dimensions without any backtesting on real financial data or implementation-ready code.

Research Flowchart

  flowchart TD
  A["Research Goal<br>Solve High-Dim Nonlinear BSDEs"] --> B["Key Methodology<br>Differential Deep Learning + Malliavin Calculus"]
  B --> C["Compute System<br>Y, Z, Γ triple via Euler-Maruyama"]
  C --> D["Data Input<br>High-dimensional financial parameters"]
  D --> E["Computational Process<br>DNN Training with Backward Optimization"]
  E --> F["Key Findings<br>Efficient Γ estimation & error convergence"]

Cited References (71)

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3. Option Pricing with Differential Interest Rates (1995) — link
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5. A numerical scheme for BSDEs (2004) — link
6. A regression-based Monte Carlo method to solve backward stochastic differential equations (2005) — link
7. Rate of convergence of an empirical regression method for solving generalized backward stochastic differential equations (2006) — link
8. A New Kind of Accurate Numerical Method for Backward Stochastic Differential Equations (2006) — link

0 of 71 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.