mlsanityData to 14 Oct 2026

A deep solver for backward stochastic Volterra integral equations

arXiv 2505.18297 ↗First posted 23 May 2025math.NA cs.LG cs.NAcited by 1

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Abstract (arXiv)

We present the first deep-learning solver for backward stochastic Volterra integral equations (BSVIEs) and their fully-coupled forward-backward variants. The method trains a neural network to approximate the two solution fields in a single stage, avoiding the use of nested time-stepping cycles that limit classical algorithms. For the decoupled case we prove a non-asymptotic error bound composed of an a posteriori residual plus the familiar square root dependence on the time step. Numerical experiments are consistent with this rate and reveal two key properties: scalability, in the sense that accuracy remains stable from low dimension up to 500 spatial variables while GPU batching keeps wall-clock time nearly constant; and generality, since the same method handles coupled systems whose forward dynamics depend on the backward solution. These results open practical access to a family of high-dimensional, time-inconsistent problems in stochastic control and quantitative finance.

Key facts. Cited by 1 paper in the map (data through 14 Oct 2026). Builds on 21 of its 51 references that are in the map. First posted on arXiv 23 May 2025.

Builds on

21 of 51 references.

References · 21

  1. 2017Solving high-dimensional partial differential equations using deep learningcited by 556
  2. 2017Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equationscited by 65
  3. 2020An overview on deep learning-based approximation methods for partial differential equationscited by 65
  4. 2016Deep Learning Approximation for Stochastic Control Problemscited by 55
  5. 2018Forward-Backward Stochastic Neural Networks: Deep Learning of High-dimensional Partial Differential Equationscited by 55
  6. 2018A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equationscited by 55
  7. 2018Convergence of the Deep BSDE Method for Coupled FBSDEscited by 43
  8. 2018Solving the Kolmogorov PDE by means of deep learningcited by 41
  9. 2018A proof that deep artificial neural networks overcome the curse of dimensionality in the numerical approximation of Kolmogorov partial differential equations with constant diffusion and nonlinear drift coefficientscited by 31
  10. 2018Machine Learning for semi linear PDEscited by 28
  11. 2020Approximation error analysis of some deep backward schemes for nonlinear PDEscited by 21
  12. 2019Three algorithms for solving high-dimensional fully-coupled FBSDEs through deep learningcited by 15
9 more
  1. 2022Convergence of a robust deep FBSDE method for stochastic controlcited by 12
  2. 2022A deep solver for BSDEs with jumpscited by 9
  3. 2020A posteriori error estimates for fully coupled McKean-Vlasov forward-backward SDEscited by 4
  4. 2024A backward differential deep learning-based algorithm for solving high-dimensional nonlinear backward stochastic differential equationscited by 4
  5. 2025Convergence of a Deep BSDE solver with jumpscited by 4
  6. 2025The deep multi-FBSDE method: a robust deep learning method for coupled FBSDEscited by 4
  7. 2024Generalized convergence of the deep BSDE method: a step towards fully-coupled FBSDEs and applications in stochastic controlcited by 2
  8. 2019Time-Inconsistent Stochastic Optimal Control Problems and Backward Stochastic Volterra Integral Equationscited by 1
  9. 2022Optimal Controls for Forward-Backward Stochastic Differential Equations: Time-Inconsistency and Time-Consistent Solutionscited by 1

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Citing papers per year: 2025: 1.

1 paper.

  1. 2025Error analysis for the deep Kolmogorov methodcited by 0