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.

Format:

Book

Author/Creator:

Grohs, Philipp, author.

Hornung, Fabian, author.

Jentzen, Arnulf, author.

Von Wurstemberger, Philippe, author.

Series:

Memoirs of the American Mathematical Society ; no. 1410.

Memoirs of the American Mathematical Society, 0065-9266 ; number 1410

Language:

English

Subjects (All):

Differential equations, Partial--Numerical solutions.

Differential equations, Partial.

Stochastic differential equations.

Approximation theory.

Neural networks (Computer science).

Options (Finance)--Prices--Mathematical models.

Options (Finance).

Machine learning.

Numerical analysis.

Probabilities.

Physical Description:

v, 93 pages ; 26 cm.

Other Title:

Correct title: Proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations

Place of Publication:

Providence, RI : AMS, American Mathematical Society, 2023.

Summary:

"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.

Contents:

Probabilistic and analytic preliminaries

Artificial neural network approximations

Artificial neural network approximations for Black-Scholes partial differential equations.

Notes:

Includes bibliographical references.

Other Format:

A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of black-scholes partial differential equations 978-1-4704-7448-5

ISBN:

9781470456320

147045632X

OCLC:

1378080225