Deep neural network expressivity for optimal stopping problems

被引:1
|
作者
Gonon, Lukas [1 ]
机构
[1] Imperial Coll London, Dept Math, London SW71NE, England
关键词
Deep neural network; Optimal stopping problem; Markov process; Expression rate; Approximation error bound; Curse of dimensionality; C45; C63; C41; AMERICAN OPTIONS; CONVERGENCE; SIMULATION; BOUNDS; APPROXIMATION; ALGORITHMS;
D O I
10.1007/s00780-024-00538-0
中图分类号
F8 [财政、金融];
学科分类号
0202 ;
摘要
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 epsilon \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\varepsilon $\end{document} by a deep ReLU neural network of size at most kappa d q epsilon - r \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\kappa d<^>{\mathfrak{q}} \varepsilon <^>{-\mathfrak{r}}$\end{document} . The constants kappa , q , r >= 0 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\kappa ,\mathfrak{q},\mathfrak{r} \geq 0$\end{document} do not depend on the dimension d \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$d$\end{document} of the state space or the approximation accuracy epsilon \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\varepsilon $\end{document} . 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 L & eacute;vy 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.
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页码:865 / 910
页数:46
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