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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Hlavní autoři: Grohs, Philipp, Hornung, Fabian, Jentzen, Arnulf, von Wurstemberger, Philippe
Médium: E-kniha Kniha
Jazyk:angličtina
Vydáno: Providence, Rhode Island American Mathematical Society 2023
Vydání:1
Edice:Memoirs of the American Mathematical Society
Témata:
Approximation theory
Differential equations, Partial-Numerical solutions
Neural networks (Computer science)
Stochastic differential equations
ISBN:147045632X, 9781470456320
ISSN:0065-9266, 1947-6221
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Obsah:
  • 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