# DeepMartingale: Duality of the Optimal Stopping Problem with Expressivity and High-Dimensional Hedging

arXiv: https://arxiv.org/abs/2510.13868 · HTML: https://mlsanity.com/paper/2510.13868/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 40 references that are in the map.
- First posted on arXiv 13 Oct 2025.

## Builds on (most cited first, 11 of 11)

- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations](https://mlsanity.com/paper/1809.02362/) (2018)
- [Approximation Bounds for Random Neural Networks and Reservoir Systems](https://mlsanity.com/paper/2002.05933/) (2020)
- [Understanding neural networks with reproducing kernel Banach spaces](https://mlsanity.com/paper/2109.09710/) (2021)
- [Optimal Stopping via Randomized Neural Networks](https://mlsanity.com/paper/2104.13669/) (2021)
- [Deep neural network expressivity for optimal stopping problems](https://mlsanity.com/paper/2210.10443/) (2022)
- [Neural reproducing kernel Banach spaces and representer theorems for deep networks](https://mlsanity.com/paper/2403.08750/) (2024)
- [Deep neural network approximation for high-dimensional parabolic Hamilton-Jacobi-Bellman equations](https://mlsanity.com/paper/2103.05744/) (2021)
- [A deep primal-dual BSDE method for optimal stopping problems](https://mlsanity.com/paper/2409.06937/) (2024)
- [Simultaneous upper and lower bounds of American-style option prices with hedging via neural networks](https://mlsanity.com/paper/2302.12439/) (2023)
- [Simulation-based Value-at-Risk for Nonlinear Portfolios](https://mlsanity.com/paper/1904.09088/) (2019)

## Built on by (most cited first, 1 of 1)

- [Dual Approaches to Stochastic Control via SPDEs and the Pathwise Hopf Formula](https://mlsanity.com/paper/2604.08155/) (2026)

## Abstract (arXiv)

We propose textitDeepMartingale, a deep-learning framework for the dual formulation of discrete-monitoring optimal stopping problems under continuous-time models. Leveraging a martingale representation, our method implements a pure-dual procedure that directly optimizes over a parameterized class of martingales, producing computable and tight dual upper bounds for the value function in high-dimensional settings without requiring any primal information or Snell-envelope approximation. We prove convergence of the resulting upper bounds under mild assumptions for both first- and second-moment losses. A key contribution is an expressivity theorem showing that textitDeepMartingale can approximate the true value function to any prescribed accuracy varepsilon using neural networks of size at most tildec d^tildeqvarepsilon^-tilder, with constants independent of the dimension d and accuracy varepsilon, thereby avoiding the curse of dimensionality. Since expressivity in this setting translates into scalability, our theory also motivates estimating the dimension scaling law to guide architecture design and the training setup in deep learning-based numerical computation and the choice of rebalancing frequency for the related hedging strategy. The learned martingale representation further yields a practical and dimension-scalable deep delta hedging strategy. Numerical experiments on high-dimensional Bermudan option benchmarks confirm convergence, expressivity, scalable training, and the stability of the resulting upper bounds and hedging performance.