  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.

Probability;Machine Learning;Numerical Analysis;Numerical Analysis;Mathematical Finance