Deep BSVIEs Parametrization and Learning-Based Applications

TL;DR

This work develops a numerical framework for backward stochastic Volterra integral equations (BSVIEs) and their reflected variants, focusing on Type-I BSVIEs with memory and time-inconsistent dynamics. It introduces a two-time, parametrized BSDE representation and proves joint measurability in the product space via Stricker–Yor results, enabling a neural-network based solver that enforces the two-time structure through backward dynamic programming. The DeepBSVIE algorithm discretizes the time interval, learns both $Y(t)$ and the two-argument kernel $Z(t,s)$ with dedicated neural networks, and provides a convergence analysis that couples discretization error with neural approximation error, extending prior deep-BSDE results to BSVIEs. The method is validated on several recursive-utility and asset-valuation applications, including memory- and ambiguity-aware utilities, exponential-growth valuations, nonlinear wealth effects, and reflected (RBSVIE) variants for regret-aversion floors, demonstrating accuracy and scalability in high dimensions with practical implications for non-Markovian finance and control problems.

Abstract

We study the numerical approximation of backward stochastic Volterra integral equations (BSVIEs) and their reflected extensions, which naturally arise in problems with time inconsistency, path dependent preferences, and recursive utilities with memory. These equations generalize classical BSDEs by involving two dimensional time structures and more intricate dependencies. We begin by developing a well posedness and measurability framework for BSVIEs in product probability spaces. Our approach relies on a representation of the solution as a parametrized family of backward stochastic equations indexed by the initial time, and draws on results of Stricker and Yor to ensure that the two parameter solution is well defined in a joint measurable sense. We then introduce a discrete time learning scheme based on a recursive backward representation of the BSVIE, combining the discretization of Hamaguchi and Taguchi with deep neural networks. A detailed convergence analysis is provided, generalizing the framework of deep BSDE solvers to the two dimensional BSVIE setting. Finally, we extend the solver to reflected BSVIEs, motivated by applications in delayed recursive utility with lower constraints.

Figures (17)

• Figure 1: Surface plots and slices of the kernel functions $Z(t,s)$. Left: Piecewise smooth kernel; Middle: Singular power law kernel $|t-s|^\alpha$; Right: Oscillatory kernel $\sin(1/(t-s))$.
• Figure 2: Comparison of the learned $Y(t)$ (red) with the analytical solution (dashed blue).
• Figure 3: Visualization of the structure of the first component of the learned kernel $Z_{i,j}$ and its continuous counterpart $Z(t,s)$.
• Figure 4: Algorithm loss across iterations evaluated at selected time steps.
• Figure 5: Visualization of estimation errors.

Theorems & Definitions (19)

• Lemma 3.1: Measurable Conditional Expectations
• Theorem 3.2: Stricker-Yor Measurability for Parameter Dependent Integrals
• Theorem 3.3: Measurability of the BSVIE Solution
• proof
• Remark 3.4
• Theorem 3.5: Existence and Uniqueness in Product Space
• proof : Proof Sketch
• Remark 3.6
• Remark 3.7: Multidimensional Extension
• Remark 5.3