# Solving high-dimensional optimal stopping problems using deep learning

arXiv: https://arxiv.org/abs/1908.01602 · HTML: https://mlsanity.com/paper/1908.01602/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Aug 2019.

## Built on by (most cited first, 22 of 22)

- [Tackling the Curse of Dimensionality with Physics-Informed Neural Networks](https://mlsanity.com/paper/2307.12306/) (2023)
- [Hutchinson Trace Estimation for High-Dimensional and High-Order Physics-Informed Neural Networks](https://mlsanity.com/paper/2312.14499/) (2023)
- [Deep neural network expressivity for optimal stopping problems](https://mlsanity.com/paper/2210.10443/) (2022)
- [Learning to Optimally Stop Diffusion Processes, with Financial Applications](https://mlsanity.com/paper/2408.09242/) (2024)
- [Deep Signature Algorithm for Multi-dimensional Path-Dependent Options](https://mlsanity.com/paper/2211.11691/) (2022)
- [Solving the optimal stopping problem with reinforcement learning: an application in financial option exercise](https://mlsanity.com/paper/2208.00765/) (2022)
- [Machine learning for option pricing: an empirical investigation of network architectures](https://mlsanity.com/paper/2307.07657/) (2023)
- [Deep Neural Operator Learning for Probabilistic Models](https://mlsanity.com/paper/2511.07235/) (2025)
- [Optimal Stopping with Gaussian Processes](https://mlsanity.com/paper/2209.14738/) (2022)
- [Regret-Optimal Federated Transfer Learning for Kernel Regression with Applications in American Option Pricing](https://mlsanity.com/paper/2309.04557/) (2023)
- [Deep Optimal Timing Strategies for Time Series](https://mlsanity.com/paper/2310.05479/) (2023)
- [A Deep-Genetic Algorithm (Deep-GA) Approach for High-Dimensional Nonlinear Parabolic Partial Differential Equations](https://mlsanity.com/paper/2311.11558/) (2023)
- [PADAM: Parallel averaged Adam reduces the error for stochastic optimization in scientific machine learning](https://mlsanity.com/paper/2505.22085/) (2025)
- [Deep Learning for the Multiple Optimal Stopping Problem](https://mlsanity.com/paper/2512.22961/) (2025)
- [On the Impact of Feeding Cost Risk in Aquaculture Valuation and Decision Making](https://mlsanity.com/paper/2309.02970/) (2023)
- [Fractional-Boundary-Regularized Deep Galerkin Method for Variational Inequalities in Mixed Optimal Stopping and Control](https://mlsanity.com/paper/2505.19309/) (2025)
- [Time Deep Gradient Flow Method for pricing American options](https://mlsanity.com/paper/2507.17606/) (2025)
- [Robust Exploratory Stopping under Ambiguity in Reinforcement Learning](https://mlsanity.com/paper/2510.10260/) (2025)
- [Reinforcement Learning for Speculative Trading under Exploratory Framework](https://mlsanity.com/paper/2604.02035/) (2026)
- [Continuous-time Optimal Stopping through Deep Reinforcement Learning](https://mlsanity.com/paper/2606.17545/) (2026)
- [Parameterized Representations via Implicit Stochastic Modulation for High-Dimensional and High-Order Neural PDE Solvers](https://mlsanity.com/paper/2606.22150/) (2026)
- [CC-AOS: Cost- and Horizon-Conditioned Amortized Backward Induction for Finite-Horizon Optimal Stopping](https://mlsanity.com/paper/2607.22774/) (2026)

## Abstract (arXiv)

Nowadays many financial derivatives, such as American or Bermudan options, are of early exercise type. Often the pricing of early exercise options gives rise to high-dimensional optimal stopping problems, since the dimension corresponds to the number of underlying assets. High-dimensional optimal stopping problems are, however, notoriously difficult to solve due to the well-known curse of dimensionality. In this work, we propose an algorithm for solving such problems, which is based on deep learning and computes, in the context of early exercise option pricing, both approximations of an optimal exercise strategy and the price of the considered option. The proposed algorithm can also be applied to optimal stopping problems that arise in other areas where the underlying stochastic process can be efficiently simulated. We present numerical results for a large number of example problems, which include the pricing of many high-dimensional American and Bermudan options, such as Bermudan max-call options in up to 5000 dimensions. Most of the obtained results are compared to reference values computed by exploiting the specific problem design or, where available, to reference values from the literature. These numerical results suggest that the proposed algorithm is highly effective in the case of many underlyings, in terms of both accuracy and speed.