\contentsline {part}{I\hspace {1em}On Convergence of Brownian Motion Monte Carlo}{4}{part.1}%
\contentsline {chapter}{\numberline {1}Introduction.}{5}{chapter.1}%
\contentsline {section}{\numberline {1.1}Notation, Definitions \& Basic notions.}{5}{section.1.1}%
\contentsline {subsection}{\numberline {1.1.1}Norms and Inner Product}{5}{subsection.1.1.1}%
\contentsline {subsection}{\numberline {1.1.2}Probability Space and Brownian Motion}{6}{subsection.1.1.2}%
\contentsline {subsection}{\numberline {1.1.3}Lipschitz and Related Notions}{9}{subsection.1.1.3}%
\contentsline {subsection}{\numberline {1.1.4}Kolmogorov Equations}{10}{subsection.1.1.4}%
\contentsline {subsection}{\numberline {1.1.5}Linear Algebra Notation and Definitions}{12}{subsection.1.1.5}%
\contentsline {subsection}{\numberline {1.1.6}$O$-type notation and function growth}{13}{subsection.1.1.6}%
\contentsline {subsection}{\numberline {1.1.7}The Iverson Bracket}{15}{subsection.1.1.7}%
\contentsline {chapter}{\numberline {2}Brownian Motion Monte Carlo}{16}{chapter.2}%
\contentsline {section}{\numberline {2.1}Brownian Motion Preliminaries}{16}{section.2.1}%
\contentsline {section}{\numberline {2.2}Monte Carlo Approximations}{20}{section.2.2}%
\contentsline {section}{\numberline {2.3}Bounds and Covnvergence}{21}{section.2.3}%
\contentsline {chapter}{\numberline {3}That $u$ is a viscosity solution}{30}{chapter.3}%
\contentsline {section}{\numberline {3.1}Some Preliminaries}{30}{section.3.1}%
\contentsline {section}{\numberline {3.2}Viscosity Solutions}{34}{section.3.2}%
\contentsline {section}{\numberline {3.3}Solutions, characterization, and computational bounds to the Kolmogorov backward equations}{53}{section.3.3}%
\contentsline {chapter}{\numberline {4}Brownian motion Monte Carlo of the non-linear case}{59}{chapter.4}%
\contentsline {part}{II\hspace {1em}A Structural Description of Artificial Neural Networks}{61}{part.2}%
\contentsline {chapter}{\numberline {5}Introduction and Basic Notions about Neural Networks}{62}{chapter.5}%
\contentsline {section}{\numberline {5.1}The Basic Definition of ANNs}{62}{section.5.1}%
\contentsline {section}{\numberline {5.2}Composition and extensions of ANNs}{66}{section.5.2}%
\contentsline {subsection}{\numberline {5.2.1}Composition}{66}{subsection.5.2.1}%
\contentsline {subsection}{\numberline {5.2.2}Extensions}{68}{subsection.5.2.2}%
\contentsline {section}{\numberline {5.3}Parallelization of ANNs}{68}{section.5.3}%
\contentsline {section}{\numberline {5.4}Affine Linear Transformations as ANNs}{72}{section.5.4}%
\contentsline {section}{\numberline {5.5}Sums of ANNs}{75}{section.5.5}%
\contentsline {subsection}{\numberline {5.5.1}Neural Network Sum Properties}{76}{subsection.5.5.1}%
\contentsline {section}{\numberline {5.6}Linear Combinations of ANNs}{83}{section.5.6}%
\contentsline {section}{\numberline {5.7}Neural Network Diagrams}{93}{section.5.7}%
\contentsline {chapter}{\numberline {6}ANN Product Approximations}{95}{chapter.6}%
\contentsline {section}{\numberline {6.1}Approximation for simple products}{95}{section.6.1}%
\contentsline {subsection}{\numberline {6.1.1}The $\prd $ network}{106}{subsection.6.1.1}%
\contentsline {section}{\numberline {6.2}Higher Approximations}{111}{section.6.2}%
\contentsline {subsection}{\numberline {6.2.1}The $\tun $ Neural Network}{112}{subsection.6.2.1}%
\contentsline {subsection}{\numberline {6.2.2}The $\pwr $ Neural Networks}{114}{subsection.6.2.2}%
\contentsline {subsection}{\numberline {6.2.3}The $\tay $ neural network}{123}{subsection.6.2.3}%
\contentsline {subsection}{\numberline {6.2.4}Neural network approximations for $e^x$.}{128}{subsection.6.2.4}%
\contentsline {chapter}{\numberline {7}A modified Multi-Level Picard and associated neural network}{129}{chapter.7}%
\contentsline {chapter}{\numberline {8}Some categorical ideas about neural networks}{132}{chapter.8}%
\contentsline {chapter}{\numberline {9}ANN first approximations}{133}{chapter.9}%
\contentsline {section}{\numberline {9.1}Activation Function as Neural Networks}{133}{section.9.1}%
\contentsline {section}{\numberline {9.2}ANN Representations for One-Dimensional Identity}{134}{section.9.2}%
\contentsline {section}{\numberline {9.3}Modulus of Continuity}{142}{section.9.3}%
\contentsline {section}{\numberline {9.4}Linear Interpolation of real-valued functions}{143}{section.9.4}%
\contentsline {subsection}{\numberline {9.4.1}The Linear Interpolation Operator}{143}{subsection.9.4.1}%
\contentsline {subsection}{\numberline {9.4.2}Neural Networks to approximate the $\lin $ operator}{144}{subsection.9.4.2}%
\contentsline {section}{\numberline {9.5}Neural network approximation of 1-dimensional functions.}{148}{section.9.5}%
\contentsline {section}{\numberline {9.6}$\trp ^h$ and neural network approximations for the trapezoidal rule.}{151}{section.9.6}%
\contentsline {section}{\numberline {9.7}Linear interpolation for multi-dimensional functions}{154}{section.9.7}%
\contentsline {subsection}{\numberline {9.7.1}The $\nrm ^d_1$ and $\mxm ^d$ networks}{154}{subsection.9.7.1}%
\contentsline {subsection}{\numberline {9.7.2}The $\mxm ^d$ neural network and maximum convolutions }{160}{subsection.9.7.2}%
\contentsline {subsection}{\numberline {9.7.3}Lipschitz function approximations}{164}{subsection.9.7.3}%
\contentsline {subsection}{\numberline {9.7.4}Explicit ANN approximations }{167}{subsection.9.7.4}%
\contentsline {part}{III\hspace {1em}A deep-learning solution for $u$ and Brownian motions}{169}{part.3}%
\contentsline {chapter}{\numberline {10}ANN representations of Brownian Motion Monte Carlo}{170}{chapter.10}%
\contentsline {chapter}{Appendices}{180}{section*.3}%