Deep neural network expressivity for optimal stopping problems

Lukas Gonon ()
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Lukas Gonon: Imperial College London

Finance and Stochastics, 2024, vol. 28, issue 3, No 6, 865-910

Abstract: 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 ε $\varepsilon $ by a deep ReLU neural network of size at most κ d q ε − r $\kappa d^{\mathfrak{q}} \varepsilon ^{-\mathfrak{r}}$ . The constants κ , q , r ≥ 0 $\kappa ,\mathfrak{q},\mathfrak{r} \geq 0$ do not depend on the dimension d $d$ of the state space or the approximation accuracy ε $\varepsilon $ . This proves that deep neural networks do not suffer from the curse of dimensionality when employed to approximate solutions of 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.

Keywords: Deep neural network; Optimal stopping problem; Markov process; Expression rate; Approximation error bound; Curse of dimensionality; 60G40; 68T07; 62M45; 91G20; 60J05 (search for similar items in EconPapers)
JEL-codes: C41 C45 C63 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s00780-024-00538-0

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