A deep solver for backward stochastic Volterra integral equations

TL;DR

This paper introduces the first deep-learning method for solving backward stochastic Volterra integral equations, enabling scalable, high-dimensional, and fully-coupled problem solutions in stochastic control and finance.

Contribution

The paper presents a novel neural network approach that efficiently solves BSVIEs without nested time steps, applicable to high-dimensional and coupled systems.

Findings

Method achieves stable accuracy up to 500 dimensions.

Error bound combines residual and time step dependence.

Approach handles coupled forward-backward systems.

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 are consistent with 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, time-inconsistent problems in stochastic control and quantitative finance.