A deep solver for backward stochastic Volterra integral equations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915562564616192 |
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| author | Andersson, Kristoffer Gnoatto, Alessandro Trillos, Camilo Andrés García |
| author_facet | Andersson, Kristoffer Gnoatto, Alessandro Trillos, Camilo Andrés García |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_18297 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A deep solver for backward stochastic Volterra integral equations Andersson, Kristoffer Gnoatto, Alessandro Trillos, Camilo Andrés García Numerical Analysis Machine Learning Probability Mathematical Finance 65C30, 60H20, 60H35, 68T07 G.1.9; G.3; I.2.6; F.2.1 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. |
| title | A deep solver for backward stochastic Volterra integral equations |
| topic | Numerical Analysis Machine Learning Probability Mathematical Finance 65C30, 60H20, 60H35, 68T07 G.1.9; G.3; I.2.6; F.2.1 |
| url | https://arxiv.org/abs/2505.18297 |