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010 a| 2023015044;

020 a| 9781470456320q| (paperback);

020 z| 9781470474485q| (pdf);

035 a| (WaSeSS)ssj0002865429;

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100 1 a| Grohs, Philipp.?| UNAUTHORIZED;

245 12 a| A proof that aritificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equationsh| [electronic resource] /c| Philipp Grohs, Fabian Hornung, Arnulf Jentzen, Philippe von Wurstemberger.;

246 1 i| Correct title:a| Proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations;

260 a| Providence, RI :b| AMS, American Mathematical Society,c| 2023.;

300 a| v, 93 pages ;c| 26 cm;

490 0 a| Memoirs of the American Mathematical Society,x| 0065-9266 ;v| volume 284, number 1410;

504 a| Includes bibliographical references (pages 89-93);

505 0 a| Probabilistic and analytic preliminaries -- Artificial neural network approximations -- Artificial neural network approximations for Black-Scholes partial differential equations.;

506 a| Available only to authorized users.;

520 a| "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 [error term] [greater than] 0 and the PDE dimension d [element of] N. We thereby prove, for the first time, that ANNs do indeed overcome the curse of dimensionality in the numerical approximation of Black-Scholes PDEs"--c| Provided by publisher.;

538 a| Mode of access: World Wide Web;

650 0 a| Differential equations, Partialx| Numerical solutions.=| ^A21574;

650 0 a| Stochastic differential equations.=| ^A69732;

650 0 a| Approximation theory.=| ^A16000;

650 0 a| Neural networks (Computer science)=| ^A263002;

650 0 a| Options (Finance)x| Pricesx| Mathematical models.=| ^A1006257;

650 7 a| Numerical analysis -- Numerical approximation and computational geometry (primarily algorithms) -- None of the above, but in this section.2| msc;

650 7 a| Partial differential equations -- Parabolic equations and systems.2| msc;

650 7 a| Probability theory and stochastic processes -- Stochastic analysis.2| msc;

655 0 a| Electronic books.=| ^A491897;

700 1 a| Hornung, Fabian.?| UNAUTHORIZED;

700 1 a| Jentzen, Arnulf.=| ^A1082204;

700 1 a| Von Wurstemberger, Philippe.?| UNAUTHORIZED;

856 40 z| Full text available from Ebook Central - Academic Completeu| https://ebookcentral.proquest.com/lib/eastcarolina/detail.action?docID=30539170;

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A proof that aritificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations / Philipp Grohs, Fabian Hornung, Arnulf Jentzen, Philippe von Wurstemberger.

Author/creator	Grohs, Philipp
Other author	Hornung, Fabian.
Other author	Jentzen, Arnulf.
Other author	Von Wurstemberger, Philippe.
Format	Electronic
Publication Info	Providence, RI : AMS, American Mathematical Society, 2023.
Description	v, 93 pages ; 26 cm
Supplemental Content	Full text available from Ebook Central - Academic Complete
Subjects	Differential equations, Partial--Numerical solutions. Stochastic differential equations. Approximation theory. Neural networks (Computer science) Options (Finance)--Prices. --Mathematical models.

Variant title	Correct title: Proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations
Series	Memoirs of the American Mathematical Society, 0065-9266 ; volume 284, number 1410
Contents	Probabilistic and analytic preliminaries -- Artificial neural network approximations -- Artificial neural network approximations for Black-Scholes partial differential equations.
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 [error term] [greater than] 0 and the PDE dimension d [element of] N. We thereby prove, for the first time, that ANNs do indeed overcome the curse of dimensionality in the numerical approximation of Black-Scholes PDEs"-- Provided by publisher.
Bibliography note	Includes bibliographical references (pages 89-93)
Access restriction	Available only to authorized users.
Technical details	Mode of access: World Wide Web
Genre/form	Electronic books.
LCCN	2023015044
ISBN	9781470456320 (paperback)
ISBN	(pdf)

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A proof that aritificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations / Philipp Grohs, Fabian Hornung, Arnulf Jentzen, Philippe von Wurstemberger.

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