mlsanityData to 14 Oct 2026

Deep neural network expressivity for optimal stopping problems

arXiv 2210.10443 ↗First posted 19 Oct 2022math.PR cs.LG cs.NAcited by 9

Not on the map. Why?

Abstract (arXiv)

This article studies deep neural network expression rates for optimal stopping problems of discrete-time Markov processes on high-dimensional state spaces. A general framework is established in which the value function and continuation value of an optimal stopping problem can be approximated with error at most varepsilon by a deep ReLU neural network of size at most kappa d^mathfrakq varepsilon^-mathfrakr. The constants kappa,mathfrakq,mathfrakr ≥ 0 do not depend on the dimension d of the state space or the approximation accuracy varepsilon. This proves that deep neural networks do not suffer from the curse of dimensionality when employed to solve optimal stopping problems. The framework covers, for example, exponential L'evy models, discrete diffusion processes and their running minima and maxima. These results mathematically justify the use of deep neural networks for numerically solving optimal stopping problems and pricing American options in high dimensions.

Key facts. Cited by 9 papers in the map (data through 14 Oct 2026). Builds on 22 of its 61 references that are in the map. First posted on arXiv 19 Oct 2022.

Builds on

22 of 61 references.

References · 22

  1. 2017DGM: A deep learning algorithm for solving partial differential equationscited by 678
  2. 2016Error bounds for approximations with deep ReLU networkscited by 574
  3. 2017Optimal approximation of piecewise smooth functions using deep ReLU neural networkscited by 179
  4. 2018Deep Hedgingcited by 108
  5. 2019Error bounds for approximations with deep ReLU neural networks in Ws,p normscited by 95
  6. 2020An overview on deep learning-based approximation methods for partial differential equationscited by 65
  7. 2015Provable approximation properties for deep neural networkscited by 60
  8. 2018A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equationscited by 55
  9. 2018Analysis of the Generalization Error: Empirical Risk Minimization over Deep Artificial Neural Networks Overcomes the Curse of Dimensionality in the Numerical Approximation of Black-Scholes Partial Differential Equationscited by 45
  10. 2019Neural networks for option pricing and hedging: a literature reviewcited by 40
  11. 2020Deep learning of free boundary and Stefan problemscited by 35
  12. 2021Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionalitycited by 34
10 more
  1. 2020Expressivity of Deep Neural Networkscited by 32
  2. 2018A proof that deep artificial neural networks overcome the curse of dimensionality in the numerical approximation of Kolmogorov partial differential equations with constant diffusion and nonlinear drift coefficientscited by 31
  3. 2020A generative adversarial network approach to calibration of local stochastic volatility modelscited by 29
  4. 2021Optimal Stopping via Randomized Neural Networkscited by 28
  5. 2021Neural networks-based algorithms for stochastic control and PDEs in financecited by 26
  6. 2019Solving high-dimensional optimal stopping problems using deep learningcited by 22
  7. 2019Rectified deep neural networks overcome the curse of dimensionality for nonsmooth value functions in zero-sum games of nonlinear stiff systemscited by 12
  8. 2019Uniform error estimates for artificial neural network approximations for heat equationscited by 11
  9. 2020Randomized optimal stopping algorithms and their convergence analysiscited by 6
  10. 2021Deep neural network approximation for high-dimensional parabolic Hamilton-Jacobi-Bellman equationscited by 5

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2022: 1; 2023: 1; 2024: 0; 2025: 5; 2026: 2.

9 papers.

  1. 2022Deep Signature Algorithm for Multi-dimensional Path-Dependent Optionscited by 6
  2. 2025Feedback-driven recurrent quantum neural network universalitycited by 5
  3. 2023Simultaneous upper and lower bounds of American-style option prices with hedging via neural networkscited by 3
  4. 2025Deep Neural Operator Learning for Probabilistic Modelscited by 3
  5. 2025DeepMartingale: Duality of the Optimal Stopping Problem with Expressivity and High-Dimensional Hedgingcited by 1
  6. 2025Deep Learning for the Multiple Optimal Stopping Problemcited by 1
  7. 2025Error analysis for the deep Kolmogorov methodcited by 0
  8. 2026Continuous-time Optimal Stopping through Deep Reinforcement Learningcited by 0
  9. 2026NeuralChaos: Optimal Adapted Approximation of Square Integrable Predictable Processescited by 0

Similar, not cited

Close in wording, no citation either way

  1. 2025Deep neural networks can provably solve Bellman equations for Markov decision processes without the curse of dimensionality
  2. 2024A time-stepping deep gradient flow method for option pricing in (rough) diffusion models
  3. 2024Solving The Dynamic Volatility Fitting Problem: A Deep Reinforcement Learning Approach
  4. 2024A deep implicit-explicit minimizing movement method for option pricing in jump-diffusion models
  5. 2024Learning to Optimally Stop Diffusion Processes, with Financial Applications
  6. 2025Deep reinforcement learning for optimal trading with partial information
  7. 2024Deep Reinforcement Learning for Online Optimal Execution Strategies
  8. 2024Low-dimensional approximations of the conditional law of Volterra processes: a non-positive curvature approach