A Proof that Artificial Neural Networks Overcome the Curse of Dimensionality in the Numerical Approximation of Black–Scholes Partial Differential Equations
Artificial neural networks (ANNs) have very successfully been used in numerical simulations for a series of computational problems ranging from image classification/image recognition, speech recognition, time series analysis, game intelligence, and computational advertising to numerical approximatio...
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| Hauptverfasser: | , , , |
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| Format: | E-Book Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Providence, Rhode Island
American Mathematical Society
2023
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| Ausgabe: | 1 |
| Schriftenreihe: | Memoirs of the American Mathematical Society |
| Schlagworte: | |
| ISBN: | 147045632X, 9781470456320 |
| ISSN: | 0065-9266, 1947-6221 |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | Artificial neural networks (ANNs) have very successfully been used in numerical simulations for a series of computational problems
ranging from image classification/image recognition, speech recognition, time series analysis, game intelligence, and computational
advertising to numerical approximations of partial differential equations (PDEs). Such numerical simulations suggest that ANNs have the
capacity to very efficiently approximate high-dimensional functions and, especially, indicate that ANNs seem to admit the fundamental
power to overcome the curse of dimensionality when approximating the high-dimensional functions appearing in the above named
computational problems. There are a series of rigorous mathematical approximation results for ANNs in the scientific literature. Some of
them prove convergence without convergence rates and some of these mathematical results even rigorously establish convergence rates but
there are only a few special cases where mathematical results can rigorously explain the empirical success of ANNs when approximating
high-dimensional functions. The key contribution of this article is to disclose that ANNs can efficiently approximate high-dimensional
functions in the case of numerical approximations of Black-Scholes PDEs. More precisely, this work reveals that the number of required
parameters of an ANN to approximate the solution of the Black-Scholes PDE grows at most polynomially in both the reciprocal of the
prescribed approximation accuracy |
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| Bibliographie: | April 2023, volume 284, number 1410 (sixth of 6 numbers) Other authors: Fabian Hornung, Arnulf Jentzen, Philippe von Wurstemberger Includes bibliographical references (p. 89-93) |
| ISBN: | 147045632X 9781470456320 |
| ISSN: | 0065-9266 1947-6221 |
| DOI: | 10.1090/memo/1410 |