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

Abstract

We propose \textit{DeepMartingale}, 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 \emph{pure-dual} procedure that directly optimizes over a parameterized class of martingales, producing computable and tight \emph{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 \textit{DeepMartingale} can approximate the true value function to any prescribed accuracy $\varepsilon$ using neural networks of size at most $\tilde{c} d^{\tilde{q}}\varepsilon^{-\tilde{r}}$, 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 \emph{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.