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prelim progress

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Shakil Rafi 2024-03-11 21:35:06 -05:00
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Dissertation/ann_first_approximations.tex
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@ -1,5 +1,5 @@
\chapter{ANN first approximations}
\section{ANN Representations for One-Dimensional Identity and some associated properties}
\section{ANN Representations for One-Dimensional Identity}
\begin{definition}[One Dimensional Identity Neural Network]\label{7.2.1}
We will denote by $\id_d \in \neu$ the neural network satisfying for all $d \in \N$ that:
@ -621,7 +621,7 @@ This completes the proof.
%\end{proof}
\section{$\trp^h$, $\etr^{n,h}$ and Neural Network Approximations For the Trapezoidal Rule.}
\begin{definition}[The $\trp$ neural network]
Let $h \in \R_{\ges 0}$. We define the $\trp^h \in \neu$ neural network as:
Let $h \in \lb 0,\infty \rp $. We define the $\trp^h \in \neu$ neural network as:
\begin{align}
\trp^h \coloneqq \aff_{\lb \frac{h}{2} \: \frac{h}{2}\rb,0}
\end{align}
@ -640,7 +640,7 @@ This completes the proof.
This a straight-forward consequence of Lemma \ref{5.3.1}
\end{proof}
\begin{definition}[The $\etr$ neural network]
Let $n\in \N$ and $h \in \R_{\ges 0}$. We define the neural network $\etr^{n,h} \in \neu$ as:
Let $n\in \N$ and $h \in \lb 0,\infty \rp$. We define the neural network $\etr^{n,h} \in \neu$ as:
\begin{align}
\etr^{n,h} \coloneqq \aff_{\underbrace{\lb \frac{h}{2} \ h \ h\ ... \ h \ \frac{h}{2}\rb}_{n+1-many},0}
\end{align}
@ -1126,7 +1126,7 @@ We will call the approximant $\max_{i \in \{0,1,\hdots, N\}}\{ f_i\}$, the \text
\end{align}
\end{proof}
\begin{remark}
We may represent the neural network diagram for $\mxm^d$ as:
We may represent the neural network diagram for $\mxm^d$ below.
\end{remark}
\begin{figure}[h]
\begin{center}

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Dissertation/ann_rep_brownian_motion_monte_carlo.tex
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@ -866,39 +866,56 @@ Let $t \in \lp 0,\infty\rp$ and $T \in \lp t,\infty\rp$. Let $\lp \Omega, \mathc
\begin{corollary}\label{cor_ues}
Let $N,n,\fn \in \N$, $h,\ve \in \lp 0,\infty\rp$, $q\in\lp 2,\infty\rp$, given $\mathsf{UES}^{N,h,q,\ve}_{n,\mathsf{G}_d, \Omega, \fn} \subsetneq \neu $, it is the case that:
\begin{align}
\E\left| \E \lb \exp\lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega } ds\rp \cdot \fu_d^T\lp \cX^{d,t,x}_{r,\Omega}\rp\rb -\frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1}\lb \exp \lp \int_t^T \alpha_d \circ \mathcal{X}^{d,t,x}_{r,\omega_i}\rp ds \cdot \fu_d^T\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rb \rb \right|\nonumber
\E\left| \E \lb \exp\lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega } ds\rp \cdot \fu_d^T\lp \cX^{d,t,x}_{r,\Omega}\rp\rb -\frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1}\lb \exp \lp \int_t^T \alpha_d \circ \mathcal{X}^{d,t,x}_{r,\omega_i} ds \rp \cdot \fu_d^T\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rb \rb \right|\nonumber
\end{align}
\end{corollary}
\begin{proof}
Note that \cite[Corollary~3.8]{hutzenthaler_strong_2021} tells us that:
\begin{align}
&\E\left| \E \lb \exp\lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega } ds\rp \cdot \fu_d^T\lp \cX^{d,t,x}_{r,\Omega}\rp\rb -\frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1}\lb \exp \lp \int_t^T \alpha_d \circ \mathcal{X}^{d,t,x}_{r,\omega_i}\rp ds \cdot \fu_d^T\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rb \rb \right|\nonumber \\
&\les \frac{\fK_p \sqrt{p-1}}{n^{\frac{1}{2}}} \lp \E \lb \left| \exp\lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega } ds\rp \cdot \fu_d^T\lp\cX^{d,t,x}_{r,\Omega}\rp \right|\rb \rp
\end{align}
Note that Taylor's theorem states that:
For the purposes of this proof let it be the case that $\fF: [0,T] \rightarrow \R$ is the function represented for all $\ft \in \lb 0,T \rb$ as:
\begin{align}
\exp\lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega}ds\rp = 1 + \int^T_t \alpha_d \circ \cX ^{d,t,x}_{r,\Omega}ds + \frac{1}{2}\lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega }\rp^2 ds + \fR_3
\ff\lp t\rp = \int^T_{T-t} \alpha_d\circ \cX^{d,t,x}_{r,\Omega} ds
\end{align}
Where $\fR_3$ is the Lagrange form of the reamainder. Thus $\exp\lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega } ds\rp \cdot u\lp T,\cX^{d,t,x}_{r,\Omega}\rp$ is rendered as:
In which case we have that $\fF\lp 0\rp = 0$, and thus we may define $u\lp t,x\rp$ as the function given by:
\begin{align}
&\exp\lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega } ds\rp \cdot \fu_d^T\lp\cX^{d,t,x}_{r,\Omega}\rp \\
&= \fu^T\lp\cX^{d,t,s}_{r,\Omega }\rp + \fu_d^T\lp \cX^{d,t,s}_{r,\Omega}\rp \cdot \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\omega } + \frac{1}{2} \fu^T \lp \cX^{d,t,x}_{r,\Omega}\rp \cdot \lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega}\rp^2 \\
&+\fR_3 \cdot \fu_d^T\lp \cX^{d,t,s}_{r,\Omega}\rp
\end{align}
\end{proof}
Jensen's Inequality, the fact that $\fu^T$ does not depend on time, and the linearity of integrals gives us:
\begin{align}
&= \fu^T\lp\cX^{d,t,s}_{r,\Omega }\rp + \fu^T\lp \cX^{d,t,s}_{r,\Omega}\rp \cdot \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\omega } ds + \frac{1}{2} \fu^T \lp \cX^{d,t,x}_{r,\Omega}\rp \cdot \lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega}ds\rp^2 \nonumber\\
&+\fR_3 \cdot \fu^T\lp \cX^{d,t,s}_{r,\Omega}\rp \nonumber\\
&\les \fu^T\lp \cX^{d,t,x}_{r,\Omega}\rp + \fu^T\lp \cX^{d,t,s}_{r,\Omega}\rp \cdot \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega} ds + \frac{1}{2}\fu^T\lp \cX^{d,t,x}_{r,\Omega}\rp\cdot \lp \frac{1}{T-t}\int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega} ds\rp^2 \\ &+ \fR_3\nonumber\\
&\les \fu^T\lp \cX^{d,t,x}_{r,\Omega}\rp + \int_t^T\fu^T\lp \cX^{d,t,x}_{r,\Omega} \rp \cdot \alpha_d \circ \cX^{d,t,x}_{r,\Omega} ds + \int^T_t \frac{1}{2\lp T-t\rp}\fu^T\lp \cX^{d,t,x}_{r,\Omega }\rp \cdot \lp \alpha_d \circ \cX^{d,t,x}_{r,\Omega }\rp^2 ds\\ &+ \fR_3\nonumber \\
&= \fu^T\lp \cX^{d,t,x}_{r,\Omega} \rp + \int^T_t \fu^T \lp \cX^{d,t,x}_{r,\Omega}\rp \cdot \alpha_d \circ \cX^{d,t,x}_{r,\Omega} + \frac{1}{2\lp T-t\rp}\fu^T\lp \cX^{d,t,x}_{r,\Omega}\rp \cdot \lp\alpha_d \circ \cX^{d,t,x}_{r,\Omega}\rp^2 ds + \fR_3\nonumber
\end{align}
Thus \cite[Lemma~2.3]{hutzenthaler_strong_2021} with $f \curvearrowleft \fu^T$ tells us that:
\begin{align}
\E
u\lp t,x\rp &= \exp \lp \ff\lp t\rp\rp \cdot \fu^T\lp \cX^{d,t,x}_{r,\Omega}\rp \nonumber\\
&= \lb \exp\lp \fF\lp 0\rp\rp + \int_0^s \ff'\lp s\rp\cdot \exp \lp \ff\lp s\rp\rp ds\rb \cdot \fu^T\lp \cX^{d,t,x}_{r,\Omega}\rp\nonumber \\
&=\fu^T\lp \cX^{d,t,x}_{r,\Omega}\rp + \int_0^s \ff'\lp s\rp \cdot \exp\lp \ff\lp s\rp\rp \cdot \fu^T\lp \cX^{d,t,x}_{r,\Omega}\rp ds \nonumber\\
&= \fu^T\lp \cX^{d,t,x}_{r,\Omega}\rp + \int^s_0 \ff'\lp s\rp\cdot u\lp s,\cX^{d,t,x}_{r,\Omega}\rp ds \nonumber \\
&=\fu^T\lp \cX^{d,t,x}\rp + \int^s_0 \fF \lp s,u\lp s,x + \cW^d_r\rp\rp
\end{align}
Then \cite[Lemma~2.3]{hutzenthaler_strong_2021} with $u \curvearrowleft u$,
% Note that Taylor's theorem states that:
% \begin{align}
% \exp\lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega}ds\rp = 1 + \int^T_t \alpha_d \circ \cX ^{d,t,x}_{r,\Omega}ds + \frac{1}{2}\lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega }\rp^2 ds + \fR_3
% \end{align}
% Where $\fR_3$ is the Lagrange form of the reamainder. Thus $\exp\lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega } ds\rp \cdot u\lp T,\cX^{d,t,x}_{r,\Omega}\rp$ is rendered as:
% \begin{align}
% &\exp\lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega } ds\rp \cdot \fu_d^T\lp\cX^{d,t,x}_{r,\Omega}\rp \\
% &= \fu^T\lp\cX^{d,t,s}_{r,\Omega }\rp + \fu_d^T\lp \cX^{d,t,s}_{r,\Omega}\rp \cdot \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\omega } + \frac{1}{2} \fu^T \lp \cX^{d,t,x}_{r,\Omega}\rp \cdot \lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega}\rp^2 \\
% &+\fR_3 \cdot \fu_d^T\lp \cX^{d,t,s}_{r,\Omega}\rp
% \end{align}
% \end{proof}
% Jensen's Inequality, the fact that $\fu^T$ does not depend on time, and the linearity of integrals gives us:
% \begin{align}
% &= \fu^T\lp\cX^{d,t,s}_{r,\Omega }\rp + \fu^T\lp \cX^{d,t,s}_{r,\Omega}\rp \cdot \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\omega } ds + \frac{1}{2} \fu^T \lp \cX^{d,t,x}_{r,\Omega}\rp \cdot \lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega}ds\rp^2 \nonumber\\
% &+\fR_3 \cdot \fu^T\lp \cX^{d,t,s}_{r,\Omega}\rp \nonumber\\
% &\les \fu^T\lp \cX^{d,t,x}_{r,\Omega}\rp + \fu^T\lp \cX^{d,t,s}_{r,\Omega}\rp \cdot \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega} ds + \frac{1}{2}\fu^T\lp \cX^{d,t,x}_{r,\Omega}\rp\cdot \lp \frac{1}{T-t}\int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega} ds\rp^2 \\ &+ \fR_3\nonumber\\
% &\les \fu^T\lp \cX^{d,t,x}_{r,\Omega}\rp + \int_t^T\fu^T\lp \cX^{d,t,x}_{r,\Omega} \rp \cdot \alpha_d \circ \cX^{d,t,x}_{r,\Omega} ds + \int^T_t \frac{1}{2\lp T-t\rp}\fu^T\lp \cX^{d,t,x}_{r,\Omega }\rp \cdot \lp \alpha_d \circ \cX^{d,t,x}_{r,\Omega }\rp^2 ds\\ &+ \fR_3\nonumber \\
% &= \fu^T\lp \cX^{d,t,x}_{r,\Omega} \rp + \int^T_t \fu^T \lp \cX^{d,t,x}_{r,\Omega}\rp \cdot \alpha_d \circ \cX^{d,t,x}_{r,\Omega} + \frac{1}{2\lp T-t\rp}\fu^T\lp \cX^{d,t,x}_{r,\Omega}\rp \cdot \lp\alpha_d \circ \cX^{d,t,x}_{r,\Omega}\rp^2 ds + \fR_3\nonumber
% \end{align}
% Thus \cite[Lemma~2.3]{hutzenthaler_strong_2021} with $f \curvearrowleft \fu^T$ tells us that:
% \begin{align}
% \E
% \end{align}
% \begin{proof}
% Note that $\fu^T$ is deterministic, and $\cX^{d,t,x}_{r,\Omega}$ is a $d$-vector of random variables, where $\mu = \mymathbb{0}_d$, and $\Sigma = \mathbb{I}_d$. Whence we have that:
% \begin{align}
@ -1038,7 +1055,7 @@ Let $t \in \lp 0,\infty\rp$ and $T \in \lp t,\infty\rp$. Let $\lp \Omega, \mathc
%\end{align}
%
%
%\end{proof}
\end{proof}

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Dissertation/main.pdf
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Dissertation/preamble.tex
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@ -2,6 +2,7 @@
\usepackage{setspace}
\doublespacing
\usepackage[toc,page]{appendix}
\usepackage{mleftright}
\usepackage{pdfpages}
\usepackage[]{amsmath}