Deep neural network expressivity for optimal stopping problems
This provides a mathematical justification for using deep neural networks to solve high-dimensional optimal stopping problems, such as pricing American options, addressing a key computational bottleneck in finance.
The paper tackles the approximation of optimal stopping problems for high-dimensional Markov processes using deep neural networks, proving that deep ReLU networks can approximate the value function with error at most ε using a network size that does not suffer from the curse of dimensionality, with constants independent of dimension and accuracy.
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 $κd^{\mathfrak{q}} \varepsilon^{-\mathfrak{r}}$. The constants $κ,\mathfrak{q},\mathfrak{r} \geq 0$ 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 Lévy 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.