A Proof That Artificial Neural Networks Overcome the Curse of Dimensionality in the Numerical Approximation of Black-Scholes Partial Differential Equations / Philipp Grohs, Fabian Hornung, and Arnulf Jentzen.

Format:

Book

Author/Creator:

Grohs, Philipp, author.

Hornung, Fabian, author.

Jentzen, Arnulf, author.

Series:

Memoirs of the American Mathematical Society ; Volume 284.

Memoirs of the American Mathematical Society Series ; Volume 284

Language:

English

Subjects (All):

Differential equations, Partial--Numerical solutions.

Differential equations, Partial.

Stochastic differential equations.

Approximation theory.

Neural networks (Computer science).

Physical Description:

1 online resource (106 pages)

Edition:

First edition.

Place of Publication:

Providence, RI : 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:

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.

Notes:

Includes bibliographical references.

Description based on print version record.

Other Format:

Print version: Grohs, Philipp A Proof That Artificial Neural Networks Overcome the Curse of Dimensionality in the Numerical Approximation of Black-Scholes Partial Differential Equations

ISBN:

9781470474485

1470474484