DeepMartingale: Duality of the Optimal Stopping Problem with Expressivity

2025-10-13

AI Summary

This paper addresses the dual formulation of optimal stopping problems under continuous-time dynamics with discrete observations, aiming to derive tight upper bounds for high-dimensional financial derivative pricing and optimal decision-making. We propose DeepMartingale, a deep learning method grounded in the martingale representation theorem: it embeds the dual optimal stopping formulation into a neural network architecture and jointly optimizes the martingale compensator and stopping policy via stochastic gradient descent. Theoretically, we establish the first expressivity guarantee for deep neural networks in this setting, proving that network size grows only polynomially in both dimension and inverse error tolerance—thereby mitigating the “curse of dimensionality.” We further provide rigorous convergence analysis and prove the tightness of the resulting upper bounds. Numerical experiments demonstrate that DeepMartingale maintains high accuracy, strong stability, and computational efficiency even in settings exceeding one hundred dimensions.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Learning to Search
• Machine Learning: Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Social networks and social learning
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Using a martingale representation, we introduce a novel deep-learning approach, which we call DeepMartingale, to study the duality of discrete-monitoring optimal stopping problems in continuous time. This approach provides a tight upper bound for the primal value function, even in high-dimensional settings. We prove that the upper bound derived from DeepMartingale converges under very mild assumptions. Even more importantly, we establish the expressivity of DeepMartingale: it approximates the true value function within any prescribed accuracy $varepsilon$ under our architectural design of neural networks whose size is bounded by $ ilde{c},D^{ ilde{q}}varepsilon^{- ilde{r}}$, where the constants $ ilde{c}, ilde{q}, ilde{r}$ are independent of the dimension $D$ and the accuracy $varepsilon$. This guarantees that DeepMartingale does not suffer from the curse of dimensionality. Numerical experiments demonstrate the practical effectiveness of DeepMartingale, confirming its convergence, expressivity, and stability.

Problem

• Solving optimal stopping problems with martingale duality
• Providing tight upper bounds in high-dimensional settings
• Overcoming curse of dimensionality through neural networks

Innovation

• DeepMartingale uses martingale representation for optimal stopping
• It provides tight upper bounds in high-dimensional settings
• The method avoids curse of dimensionality through neural networks