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A deep solver for backward stochastic Volterra integral equations

Kristoffer Andersson, Alessandro Gnoatto, Camilo Andrés García Trillos

TL;DR

The paper addresses solving high-dimensional backward stochastic Volterra integral equations (BSVIEs) and their forward–backward variants using a neural-network-based end-to-end solver. It introduces a variational, time-discrete formulation that directly parameterizes the Y and Z fields with neural networks and minimizes a residual-based loss derived from the BSVIE dynamics, avoiding nested time-stepping. A non-asymptotic error bound is established for the decoupled case, linking discretization error to the residual and demonstrating a convergence rate of order $h$. Numerical experiments across additive and multiplicative noise, as well as fully coupled FSDE-BSVIEs, validate the theory, show strong accuracy in high dimensions, and reveal practical scalability for time-inconsistent problems in stochastic control and quantitative finance.

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.

A deep solver for backward stochastic Volterra integral equations

TL;DR

The paper addresses solving high-dimensional backward stochastic Volterra integral equations (BSVIEs) and their forward–backward variants using a neural-network-based end-to-end solver. It introduces a variational, time-discrete formulation that directly parameterizes the Y and Z fields with neural networks and minimizes a residual-based loss derived from the BSVIE dynamics, avoiding nested time-stepping. A non-asymptotic error bound is established for the decoupled case, linking discretization error to the residual and demonstrating a convergence rate of order . Numerical experiments across additive and multiplicative noise, as well as fully coupled FSDE-BSVIEs, validate the theory, show strong accuracy in high dimensions, and reveal practical scalability for time-inconsistent problems in stochastic control and quantitative finance.

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.

Paper Structure

This paper contains 22 sections, 7 theorems, 76 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A decoupled FSDE-BSVIE
  4. A coupled FSDE-BSVIE
  5. Variational formulations and temporal discretization
  6. A time continuous variational formulation
  7. A time discrete variational formulation
  8. Error analysis
  9. Fully implementable scheme and neural network details
  10. Fully implementable algorithms
  11. Specification of the neural networks
  12. Numerical experiments
  13. Examples where the error analysis apply
  14. Example 1A: Additive noise
  15. Example 1B: Multiplicative noise
  16. ...and 7 more sections

Key Result

Theorem 2.1

Under assumptions ass:1-ass:2, it holds that:

Figures (10)

  • Figure 1: Comparison of the approximated $Y$ with the reference solutions for Example 1A. Left: Three representative sample paths. Right: The sample mean and the 25th and 75th percentiles.
  • Figure 2: Comparison of the approximated $Z_{t,s}$ with the reference solution for different values of $t$ for Example 1A. Left: One representative sample path of the first (of 5) component of $Z_{t,s}$. Right: A sample mean for the first component of $Z_{t,s}$.
  • Figure 3: Comparison of the approximated $Y$ with the reference solutions for Example 1B. Left: Three representative sample paths. Right: The sample mean and the 5th and 95th percentiles.
  • Figure 4: Comparison of the approximated $Z_{t,s}$ with the reference solution for different values of $t$ for Example 1B. Left: One representative sample path of the first (of 5) component of $Z_{t,s}$. Right: A sample mean the first component of $Z_{t,s}$.
  • Figure 5: Empirical convergence plot for our approximate $Y$, $Z$ and the post-optimization value of the loss function. Left: Example 1A. Right: Example 1B.
  • ...and 5 more figures

Theorems & Definitions (19)

  • Example 1.1: Social discounting, see brody2018social
  • Example 1.2
  • Theorem 2.1
  • proof
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • Theorem 4.3
  • proof
  • Theorem 4.4
  • ...and 9 more