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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Main Authors: Grohs, Philipp, Hornung, Fabian, Jentzen, Arnulf, von Wurstemberger, Philippe
Format: eBook Book
Language:English
Published: Providence, Rhode Island American Mathematical Society 2023
Edition:1
Series:Memoirs of the American Mathematical Society
Subjects:
Approximation theory
Differential equations, Partial-Numerical solutions
Neural networks (Computer science)
Stochastic differential equations
ISBN:147045632X, 9781470456320
ISSN:0065-9266, 1947-6221
Online Access:Get full text
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Abstract 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
AbstractList 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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Author Hornung, Fabian
von Wurstemberger, Philippe
Grohs, Philipp
Jentzen, Arnulf
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BackLink https://cir.nii.ac.jp/crid/1130577729740291001$$DView record in CiNii
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Notes April 2023, volume 284, number 1410 (sixth of 6 numbers)
Other authors: Fabian Hornung, Arnulf Jentzen, Philippe von Wurstemberger
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Snippet Artificial neural networks (ANNs) have very successfully been used in numerical simulations for a series of computational problems ranging from image...
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SubjectTerms Approximation theory
Differential equations, Partial-Numerical solutions
Neural networks (Computer science)
Stochastic differential equations
TableOfContents Introduction -- Probabilistic and analytic preliminaries -- Artificial neural network approximations -- Artificial neural network approximations for Black-Scholes partial differential equations
Cover -- Title page -- Chapter 1. Introduction -- Acknowlegments -- Chapter 2. Probabilistic and analytic preliminaries -- 2.1. Monte Carlo approximations -- 2.2. Properties of affine functions -- 2.3. A priori estimates for solutions of stochastic differential equations -- 2.4. Stochastic differential equations with affine coefficient functions -- 2.5. Viscosity solutions for partial differential equations -- Chapter 3. Artificial neural network approximations -- 3.1. Construction of a realization on the artificial probability space -- 3.2. Approximation error estimates -- 3.3. Cost estimates -- 3.4. Representation properties for artificial neural networks -- 3.5. Cost estimates for artificial neural networks -- 3.6. Artificial neural networks with continuous activation functions -- Chapter 4. Artificial neural network approximations for Black-Scholes partial differential equations -- 4.1. Elementary properties of the Black-Scholes model -- 4.2. Transformations of viscosity solutions -- 4.3. Artificial neural network approximations for basket call options -- 4.4. Artificial neural network approximations for basket put options -- 4.5. Artificial neural network approximations for call on max options -- 4.6. Artificial neural network approximations for call on min options -- Bibliography -- Back Cover
Title A Proof that Artificial Neural Networks Overcome the Curse of Dimensionality in the Numerical Approximation of Black–Scholes Partial Differential Equations
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