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Deep BSVIEs parametrization and learning-based applications.

Nacira Agram¹, Giulia Pucci¹

¹Department of Mathematics, KTH Royal Institute of Technology, Stockholm, 100 44, Sweden.

Neural Networks : the Official Journal of the International Neural Network Society

|February 18, 2026

Summary

This summary is machine-generated.

This study introduces a novel numerical method for backward stochastic Volterra integral equations (BSVIEs), crucial for financial modeling with memory. The deep learning approach accurately approximates solutions for these complex equations and their reflected variants.

Keywords:

BSVIEs Deep learning Neural network solvers RBSVIEs Stricker-Yor measurability

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Area of Science:

Numerical analysis
Stochastic processes
Machine learning in finance

Background:

Backward stochastic Volterra integral equations (BSVIEs) are essential for modeling complex financial scenarios like time inconsistency and path-dependent preferences.
Existing methods struggle with the two-dimensional time structure and intricate dependencies inherent in BSVIEs.

Purpose of the Study:

To develop a robust numerical approximation framework for BSVIEs and their reflected extensions.
To establish a well-posedness and measurability foundation for BSVIEs in product probability spaces.
To extend deep learning-based solvers for backward stochastic differential equations (BSDEs) to the BSVIE setting.

Main Methods:

Developed a framework for BSVIE well-posedness and measurability using a parametrized family of backward stochastic equations.
Introduced a discrete-time learning scheme combining Hamaguchi-Taguchi discretization with deep neural networks.
Generalized deep BSDE solver techniques to address the two-dimensional time structure of BSVIEs.

Main Results:

Established a rigorous convergence analysis for the proposed deep learning scheme applied to BSVIEs.
Successfully extended the numerical solver to handle reflected BSVIEs, enabling applications in areas like delayed recursive utility.
Demonstrated the effectiveness of the method in approximating solutions for complex financial models.

Conclusions:

The proposed deep learning approach provides an effective and accurate numerical method for solving BSVIEs and their reflected variants.
This work bridges the gap between theoretical BSVIE frameworks and practical computational solutions.
The findings have significant implications for quantitative finance, particularly in modeling recursive utilities and financial derivatives with memory effects.