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A deep solver for backward stochastic Volterra integral equations

Author

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  • Kristoffer Andersson
  • Alessandro Gnoatto
  • Camilo Andr'es Garc'ia Trillos

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.

Suggested Citation

  • Kristoffer Andersson & Alessandro Gnoatto & Camilo Andr'es Garc'ia Trillos, 2025. "A deep solver for backward stochastic Volterra integral equations," Papers 2505.18297, arXiv.org, revised Jul 2025.
    • Handle: RePEc:arx:papers:2505.18297
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    References listed on IDEAS

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    1. Francesca Biagini & Alessandro Gnoatto & Immacolata Oliva, 2019. "A unified approach to xVA with CSA discounting and initial margin," Papers 1905.11328, arXiv.org, revised Mar 2021.
    2. Kristoffer Andersson & Alessandro Gnoatto & Marco Patacca & Athena Picarelli, 2022. "A deep solver for BSDEs with jumps," Papers 2211.04349, arXiv.org, revised May 2025.
    3. Dorje C. Brody & Lane P. Hughston, 2018. "Social Discounting And The Long Rate Of Interest," Mathematical Finance, Wiley Blackwell, vol. 28(1), pages 306-334, January.
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