# A deep solver for backward stochastic Volterra integral equations

arXiv: https://arxiv.org/abs/2505.18297 · HTML: https://mlsanity.com/paper/2505.18297/

## Key facts

- Cited by 1 papers 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 (most cited first, 21 of 21)

- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations](https://mlsanity.com/paper/1709.05963/) (2017)
- [An overview on deep learning-based approximation methods for partial differential equations](https://mlsanity.com/paper/2012.12348/) (2020)
- [Deep Learning Approximation for Stochastic Control Problems](https://mlsanity.com/paper/1611.07422/) (2016)
- [Forward-Backward Stochastic Neural Networks: Deep Learning of High-dimensional Partial Differential Equations](https://mlsanity.com/paper/1804.07010/) (2018)
- [A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations](https://mlsanity.com/paper/1809.02362/) (2018)
- [Convergence of the Deep BSDE Method for Coupled FBSDEs](https://mlsanity.com/paper/1811.01165/) (2018)
- [Solving the Kolmogorov PDE by means of deep learning](https://mlsanity.com/paper/1806.00421/) (2018)
- [A 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 coefficients](https://mlsanity.com/paper/1809.07321/) (2018)
- [Machine Learning for semi linear PDEs](https://mlsanity.com/paper/1809.07609/) (2018)
- [Approximation error analysis of some deep backward schemes for nonlinear PDEs](https://mlsanity.com/paper/2006.01496/) (2020)
- [Three algorithms for solving high-dimensional fully-coupled FBSDEs through deep learning](https://mlsanity.com/paper/1907.05327/) (2019)
- [Convergence of a robust deep FBSDE method for stochastic control](https://mlsanity.com/paper/2201.06854/) (2022)
- [A deep solver for BSDEs with jumps](https://mlsanity.com/paper/2211.04349/) (2022)
- [A posteriori error estimates for fully coupled McKean-Vlasov forward-backward SDEs](https://mlsanity.com/paper/2007.07731/) (2020)
- [A backward differential deep learning-based algorithm for solving high-dimensional nonlinear backward stochastic differential equations](https://mlsanity.com/paper/2404.08456/) (2024)
- [Convergence of a Deep BSDE solver with jumps](https://mlsanity.com/paper/2501.09727/) (2025)
- [The deep multi-FBSDE method: a robust deep learning method for coupled FBSDEs](https://mlsanity.com/paper/2503.13193/) (2025)
- [Generalized convergence of the deep BSDE method: a step towards fully-coupled FBSDEs and applications in stochastic control](https://mlsanity.com/paper/2403.18552/) (2024)
- [Time-Inconsistent Stochastic Optimal Control Problems and Backward Stochastic Volterra Integral Equations](https://mlsanity.com/paper/1911.04995/) (2019)
- [Optimal Controls for Forward-Backward Stochastic Differential Equations: Time-Inconsistency and Time-Consistent Solutions](https://mlsanity.com/paper/2209.08994/) (2022)

## Built on by (most cited first, 1 of 1)

- [Error analysis for the deep Kolmogorov method](https://mlsanity.com/paper/2508.17167/) (2025)

## 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.