Deep Learning Methods for Backward Stochastic Volterra Integral Equations with Applications

Overview

Abstract 

Backward stochastic Volterra integral equations (BSVIEs) provide a natural framework for modeling stochastic systems with memory and path dependence. Such equations arise in several areas including recursive utilities, stochastic control, rough volatility models, and path-dependent partial differential equations. Despite their importance, numerical methods for BSVIEs remain relatively limited in the literature due to the presence of bi-temporal processes and the resulting analytical and computational challenges.

In this talk, I present a deep learning framework for approximating the solution of BSVIEs based on neural network representations of the solution components. The approach builds on recent advances in deep BSDE methods, which exploit nonlinear Feynman–Kac representations to transform high-dimensional partial differential equations into stochastic systems that can be approximated using Monte Carlo simulation and neural networks.

Extending these ideas to the Volterra setting requires learning both the value process (Y(t)) and the two-parameter component (Z(t,s)), leading to a neural architecture specifically designed for BSVIEs. I will discuss the structure of the algorithm and present convergence results for the proposed learning scheme, followed by numerical experiments illustrating the performance of the method. I will also briefly discuss extensions to reflected BSVIEs and applications to recursive utility models.