A deep solver for backward stochastic Volterra integral equations

2025-05-23

AI Summary

This work addresses backward stochastic Volterra integral equations (BSVIEs) and their fully coupled forward–backward variants in high-dimensional, path-dependent settings. Methodologically, it introduces the first end-to-end deep learning solver, employing neural networks to jointly approximate the dual solution fields within a mesh-free, non-nested temporal discretization framework, enabling efficient GPU-parallel residual minimization. Theoretically, it establishes the first non-asymptotic error bound for BSVIEs. Empirically, the solver scales robustly to dimensions up to 500, achieves stable accuracy with a square-root convergence rate in time-step size, and—uniquely—solves both decoupled and fully coupled cases using a single training run, thereby circumventing the computational bottlenecks of conventional iterative schemes. This advances the tractability and generality of high-dimensional path-dependent problems in stochastic control and quantitative finance.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Computer Vision: Learning & Optimization for CV
• Machine Learning: Neuro-Symbolic Learning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Social networks and social learning
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

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 confirm this rate and reveal two key properties: emph{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 emph{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, path-dependent problems in stochastic control and quantitative finance.

Problem

• Solves backward stochastic Volterra integral equations (BSVIEs) using deep learning
• Avoids nested time-stepping cycles in classical algorithms
• Handles high-dimensional path-dependent problems in finance and control

Innovation

• Deep-learning solver for BSVIEs
• Single-stage neural network training
• Scalable to 500 spatial variables