Deep neural network expressivity for optimal stopping problems

TL;DR

This paper proves that deep neural networks can efficiently approximate the value and continuation functions in high-dimensional optimal stopping problems without suffering from the curse of dimensionality, supporting their use in financial modeling.

Contribution

The paper establishes a general framework showing deep neural networks can approximate optimal stopping value functions with errors independent of dimension, demonstrating their effectiveness in high-dimensional settings.

Findings

Neural networks can approximate value functions with size independent of dimension.

The framework applies to models like Lévy processes and diffusions.

Results justify neural network use in high-dimensional American option pricing.

Abstract

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 $\kappa d^{\mathfrak{q}} \varepsilon^{-\mathfrak{r}}$. The constants $\kappa,\mathfrak{q},\mathfrak{r} \geq 0$ 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.