A deep solver for backward stochastic Volterra integral equations

• URL: http://arxiv.org/abs/2505.18297v2
• Date: Wed, 02 Jul 2025 07:12:03 GMT
• Title: A deep solver for backward stochastic Volterra integral equations
• Authors: Kristoffer Andersson, Alessandro Gnoatto, Camilo Andrés García Trillos,
• Abstract summary: We present the first deep-learning solver for backward Volterra integral equations (BSVIEs)<n>The method trains a neural network to approximate the two solution fields in a single stage.<n>These results open practical access to a family of high-dimensional, path-dependent problems in control and quantitative finance.
• Score: 44.99833362998488
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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