DeepMartingale: Duality of the Optimal Stopping Problem with Expressivity
Junyan Ye, Hoi Ying Wong
TL;DR
DeepMartingale introduces a martingale-representation-based deep learning framework to study the duality of discrete-monitoring optimal stopping in continuous-time models. It constructs a pure dual upper bound for the primal value $Y^*$ using Doob martingales, and proves convergence under mild assumptions. The paper also establishes expressivity: neural networks can approximate the true value within any $\varepsilon$ with size bounded by $\tilde{c}\,D^{\tilde{q}}\varepsilon^{-\tilde{r}}$, independent of dimension, thus overcoming the curse of dimensionality in this setting. Numerical experiments on Bermudan max-call and basket-put demonstrate convergence, stability, and high-dimensional effectiveness, highlighting the method's practical impact and independence from primal solutions.
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 $\tilde{c}\,D^{\tilde{q}}\varepsilon^{-\tilde{r}}$, where the constants $\tilde{c}, \tilde{q}, \tilde{r}$ 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.