From 86910c81e8452584c75acf0348dadeded4d87f68 Mon Sep 17 00:00:00 2001 From: Shakil Rafi Date: Mon, 11 Mar 2024 21:35:06 -0500 Subject: [PATCH] prelim progress --- Dissertation/ann_first_approximations.tex | 8 +-- .../ann_rep_brownian_motion_monte_carlo.tex | 61 +++++++++++------- Dissertation/main.pdf | Bin 6758921 -> 6751476 bytes Dissertation/preamble.tex | 1 + 4 files changed, 44 insertions(+), 26 deletions(-) diff --git a/Dissertation/ann_first_approximations.tex b/Dissertation/ann_first_approximations.tex index dd83cd8..a5b0f8c 100644 --- a/Dissertation/ann_first_approximations.tex +++ b/Dissertation/ann_first_approximations.tex @@ -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} diff --git a/Dissertation/ann_rep_brownian_motion_monte_carlo.tex b/Dissertation/ann_rep_brownian_motion_monte_carlo.tex index f2b2677..565f6c7 100644 --- a/Dissertation/ann_rep_brownian_motion_monte_carlo.tex +++ b/Dissertation/ann_rep_brownian_motion_monte_carlo.tex @@ -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 + 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} - \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} + 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 $T \in \lp t,\infty\rp$. 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