Deep Operator BSDE: a Numerical Scheme to Approximate Solution Operators

2026-07-09 · Source: cs.LG updates on arXiv.org · Field: Science & Research — Mathematics & Computational Sciences, Artificial Intelligence & Machine Learning, Capital Markets & Investment Management · Depth: Expert, extended

Summary

The Deep Operator BSDE, a numerical method proposed on December 4, 2024, approximates the solution operator of Backward Stochastic Differential Equations (BSDEs). Unlike traditional approaches that solve for fixed terminal conditions, this method learns the entire operator. It integrates the Wiener chaos decomposition with the classical Euler scheme for BSDEs and implements the approximation using neural networks. The scheme demonstrates convergence under mild assumptions, with a specified rate in more restrictive scenarios. Numerical examples, including dynamic risk measures, conditional g-expectations, and financial pricing/hedging of options like a discrete down and out barrier call and a discrete Asian call basket option, validate its accuracy and competitive performance against conventional methods.

Key takeaway

For quantitative analysts or machine learning engineers evaluating financial derivatives or dynamic risk measures with varying terminal conditions, this Deep Operator BSDE method offers a significant efficiency improvement. Instead of re-running computations for each new condition, you can learn the entire solution operator once. This approach is particularly beneficial in high-dimensional settings, allowing for more flexible and computationally efficient analysis of complex stochastic systems.

Key insights

This method approximates the entire solution operator of Backward Stochastic Differential Equations, enabling evaluation across varying terminal conditions.

Principles

• BSDE solution operators are globally Lipschitz.
• Wiener chaos decomposition provides a basis for L^2(F_T) functions.
• Infinite-dimensional FBSDE systems admit Markovian representations.

Method

The Operator Euler scheme recursively approximates solution operators by combining the classical Euler scheme with Wiener chaos decomposition, implemented via neural networks solving an optimization problem.

In practice

• Apply to dynamic risk measures and conditional g-expectations.
• Use for pricing and hedging complex financial derivatives.
• Neural networks handle high-dimensional Markovian map approximations.

Topics

• Backward Stochastic Differential Equations
• Numerical Methods
• Wiener Chaos Decomposition
• Neural Networks
• Financial Engineering
• Dynamic Risk Measures

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Editorial summary, takeaway, and curation by AIssential. Original article published by cs.LG updates on arXiv.org.