Deep BSVIEs parametrization and learning-based applications

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.