A deep solver for backward stochastic Volterra integral equations
This work provides practical access to high-dimensional, time-inconsistent problems in stochastic control and quantitative finance, representing a novel method for a known bottleneck.
The authors tackled the problem of solving backward stochastic Volterra integral equations (BSVIEs) and their coupled variants, which are high-dimensional and time-inconsistent, by developing the first deep-learning solver that avoids nested time-stepping and achieves scalability up to 500 spatial variables with stable accuracy and constant wall-clock time.
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.