From 8836f29605f4370c7f970ffe5fecfcb2a84c6a26 Mon Sep 17 00:00:00 2001 From: Shakil Rafi Date: Tue, 9 Apr 2024 09:47:55 -0500 Subject: [PATCH] Removed all references to learning --- .DS_Store | Bin 18436 -> 18436 bytes Dissertation/Brownian_motion_monte_carlo.tex | 36 ++++---- Dissertation/Introduction.tex | 11 ++- Dissertation/ann_first_approximations.tex | 19 +++- Dissertation/ann_product.tex | 71 +++++++++------ .../ann_rep_brownian_motion_monte_carlo.tex | 83 +++++++++++------- Dissertation/appendices.tex | 18 ++-- Dissertation/conclusions-further-research.tex | 4 +- Dissertation/front_matter.tex | 24 +++-- Dissertation/main.bib | 2 +- Dissertation/main.pdf | Bin 6821169 -> 7110360 bytes Dissertation/main.tex | 6 +- Dissertation/neural_network_introduction.tex | 46 +++++----- Dissertation/nn-example.png | Bin 0 -> 144736 bytes Dissertation/preamble.tex | 10 +-- Dissertation/u_visc_sol.tex | 5 +- MLP Ideas/MLP_ideas.pdf | Bin 177566 -> 283102 bytes MLP and DNN Material/.DS_Store | Bin 12292 -> 12292 bytes 18 files changed, 197 insertions(+), 138 deletions(-) create mode 100644 Dissertation/nn-example.png diff --git a/.DS_Store b/.DS_Store index b5a38f7718ebdfdfacb2bf2e4ad1872c4763c0de..3920604213c4f9799c964c006755bdcaa5a89c3b 100644 GIT binary patch delta 285 zcmZpfz}PZ@af7Df&oLQARIa+C#R=fazaYlZ*Cr~V)D77pzwLCh#C^IFvATc>JFJ0OG0|OW+ zI|NLAr6Z217YRAHyt$H4Hl# zjxyY0c*F3Gk%du=QHfET(S*^K(Vo$RF_1BcF_|%gF>~`~WeLv7reZ>qH%KS2Nmf@I znj20wmr|U3PE2L8u!ZL44Z0Cbo4F0N#VQ_MOZUNY0=6}yN3#fi# U+RS1xlWp^CwRFbKEDlf>00}TvHUIzs delta 386 zcmZpfz}PZ@af7BJs;iAmO>`6tYIPKKkyB1-YPIEyjeu`6B7^6R3<29ncS$ayV=$vfo=1A NwRFbKEDkdvEC7^YUIYLD diff --git a/Dissertation/Brownian_motion_monte_carlo.tex b/Dissertation/Brownian_motion_monte_carlo.tex index d5f2f40..86e7f59 100644 --- a/Dissertation/Brownian_motion_monte_carlo.tex +++ b/Dissertation/Brownian_motion_monte_carlo.tex @@ -103,7 +103,7 @@ and let $(\Omega, \mathcal{F},\mathbb{P})$ be a probability space. Let $\mathcal \end{align} and let let $U^\theta:[0,T] \times \mathbb{R}^d \times \Omega \rightarrow \mathbb{R}$, $\theta \in \Theta$ satisfy, $\theta \in \Theta$, $t \in [0,T]$, $x\in \mathbb{R}^d$, that: \begin{align}\label{(2.1.4)} - U^\theta_m(t,x) = \frac{1}{m}\left[\sum^{m}_{k=1}g\left(x+\mathcal{W}^{(\theta,0,-k)}_{T-t}\right)\right] + U^\theta_m(t,x) = \frac{1}{m}\left[\sum^{m}_{k=1}g\left(x+\mathcal{W}^{\theta}_{T-t}\right)\right] \end{align} \end{definition} \begin{lemma} \label{lemma1.1} @@ -118,7 +118,7 @@ Assume Setting \ref{primarysetting} then: \end{enumerate} \end{lemma} -\begin{proof} For (i) Consider that $\mathcal{W}^{(\theta,0,-k)}_{T-t}$ are continuous random fields and that $g\in C(\mathbb{R}^d,\mathbb{R})$, we have that $U^\theta(t,x)$ is the composition of continuous functions with $m > 0$ by hypothesis, ensuring no singularities. Thus $U^\theta: [0,T] \times \mathbb{R}^d\times \Omega \rightarrow \mathbb{R}$ is a continuous random field. +\begin{proof} For (i) Consider that $\mathcal{W}^{\theta}_{T-t}$ are continuous random fields and that $g\in C(\mathbb{R}^d,\mathbb{R})$, we have that $U^\theta(t,x)$ is the composition of continuous functions with $m > 0$ by hypothesis, ensuring no singularities. Thus $U^\theta: [0,T] \times \mathbb{R}^d\times \Omega \rightarrow \mathbb{R}$ is a continuous random field. \medskip @@ -154,7 +154,7 @@ We next claim that for all $s\in [0,T]$, $t\in[s,T]$, $\theta \in \Theta$ it hol To prove this claim observe the triangle inequality and (\ref{(2.1.4)}), demonstrate that for all $s\in[0,T]$, $t\in[s,T]$, $\theta \in \Theta$, it holds that: \begin{align}\label{(1.18)} - \mathbb{E}\lb \lv U^\theta \lp t,x+\mathcal{W}^\theta_{t-s}\rp \rv \rb \leqslant \frac{1}{m}\left[ \sum^{m}_{i=1}\mathbb{E}\lb \lv g \lp x+\mathcal{W}^\theta_{t-s}+\mathcal{W}^{(\theta,0,-i)}_{T-t} \rp \rv \rb \rb + \mathbb{E}\lb \lv U^\theta \lp t,x+\mathcal{W}^\theta_{t-s}\rp \rv \rb \leqslant \frac{1}{m}\left[ \sum^{m}_{i=1}\mathbb{E}\lb \lv g \lp x+\mathcal{W}^\theta_{t-s}+\mathcal{W}^{\theta}_{T-t} \rp \rv \rb \rb \end{align} Now observe that (\ref{(2.1.6)}) and the fact that $(W^\theta)_{\theta \in \Theta}$ are independent imply that for all $s \in [0,T]$, $t\in [s,T]$, $\theta \in \Theta$, $i\in \mathbb{Z}$ it holds that: @@ -179,11 +179,11 @@ Combining (\ref{(1.16)}), (\ref{(1.20)}), and (\ref{(1.21)}) completes the proof \begin{enumerate}[label = (\roman*)] \item it holds that $t \in [0,T],x\in \mathbb{R}^d$ that: \begin{align} - \mathbb{E}\lb \lv U^0 \lp t,x \rp \rv \rb + \mathbb{E}\lb \lv g \lp x+\mathcal{W}^{(0,0,-1)}_{T-t} \rp \rv \rb < \infty + \mathbb{E}\lb \lv U^0 \lp t,x \rp \rv \rb + \mathbb{E}\lb \lv g \lp x+\mathcal{W}^{0}_{T-t} \rp \rv \rb < \infty \end{align} \item it holds that $t\in [0,T],x\in \mathbb{R}^d$ that: \begin{align} - \mathbb{E}\lb U^0\lp t,x \rp \rb = \mathbb{E} \lb g \lp x+\mathcal{W}^{(0,0,-1)}_{T-t}\rp\rb + \mathbb{E}\lb U^0\lp t,x \rp \rb = \mathbb{E} \lb g \lp x+\mathcal{W}^{0}_{T-t}\rp\rb \end{align} \end{enumerate} \end{corollary} @@ -191,12 +191,12 @@ Combining (\ref{(1.16)}), (\ref{(1.20)}), and (\ref{(1.21)}) completes the proof \begin{proof} (i) is a restatement of Lemma \ref{lem:1.20} in that for all $t\in [0,T]$: \begin{align} - &\mathbb{E}\left[ \left| U^0\left( t,x \right) \right| \right] + \mathbb{E} \left[ \left|g \left(x+\mathcal{W}^{(0,0,-1)}_{T-t}\right)\right|\right] \nonumber\\ + &\mathbb{E}\left[ \left| U^0\left( t,x \right) \right| \right] + \mathbb{E} \left[ \left|g \left(x+\mathcal{W}^{0}_{T-t}\right)\right|\right] \nonumber\\ &<\mathbb{E} \left[ \left|U^\theta \lp t,x+\mathcal{W}^\theta_{t-s} \rp \right| \right] +\mathbb{E}\left[ \left|g \left(x+\mathcal{W}^\theta_{t-s}\right) \right| \right]+ \int^T_s \mathbb{E}\lb \lv U^\theta \lp r,x+\mathcal{W}^\theta_{r-s} \rp \rv \rb dr \nonumber\\ &< \infty \end{align} -Furthermore (ii) is a restatement of (\ref{(1.14)}) with $\theta = 0$, $m=1$, and $k=1$. This completes the proof of Corollary \ref{cor:1.20.1}. +Furthermore (ii) is a restatement of Lemma \ref{lem:1.20} with $\theta = 0$, $m=1$, and $k=1$. This completes the proof of Corollary \ref{cor:1.20.1}. \end{proof} \section{Monte Carlo Approximations} @@ -252,7 +252,7 @@ This completes the proof of the lemma. \end{proof} \section{Bounds and Covnvergence} -\begin{lemma}\label{lem:1.21} Assume Setting \ref{def:1.18}. Then it holds for all $t\in [0,T]$, $x\in \mathbb{R}^d$ +\begin{lemma}\label{lem:1.21} Assume Setting \ref{primarysetting}. Then it holds for all $t\in [0,T]$, $x\in \mathbb{R}^d$ \begin{align} &\left(\E\left[\left|U^0(t,x+\mathcal{W}^0_t)-\E \left[U^0 \left(t,x+\mathcal{W}^0_t \right)\right]\right|^\mathfrak{p}\right]\right)^{\frac{1}{\mathfrak{p}}} \nonumber\\ &\leqslant \frac{\mathfrak{m}}{m^{\frac{1}{2}}} \left[\left(\E\left[ \lv g \lp x+\mathcal{W}^0_T \rp \rv^\mathfrak{p}\right]\right)^{\frac{1}{\mathfrak{p}}}\right] @@ -264,19 +264,19 @@ This completes the proof of the lemma. G_k(t,x) = g\left(x+\mathcal{W}^{(0,0,-k)}_{T-t}\right) \end{align} \medskip -Observe that the hypothesis that $(\mathcal{W}^\theta)_{\theta \in \Theta}$ are independent Brownian motions and the hypothesis that $g \in C(\mathbb{R}^d,\mathbb{R})$ assure that for all $t \in [0,T]$,$x\in \mathbb{R}^d$ it holds that $(G_k(t,x))_{k\in \mathbb{Z}}$ are i.i.d. random variables. This and Corollary \ref{cor:1.22.2} (applied for every $t\in [0,T]$, $x\in \mathbb{R}^d$ with $p \curvearrowleft \mathfrak{p}$, $n \curvearrowleft m$, $(X_k)_{k\in \{1,2,...,m\}} \curvearrowleft (G_k(t,x))_{k\in \{1,2,...,m\}}$), with the notation of Corollary \ref{cor:1.22.2} ensure that for all $t\in [0,T]$, $x \in \mathbb{R}^d$, it holds that: +Observe that the hypothesis that $(\mathcal{W}^\theta)_{\theta \in \Theta}$ are independent Brownian motions and the hypothesis that $g \in C(\mathbb{R}^d,\mathbb{R})$ assure that for all $t \in [0,T]$,$x\in \mathbb{R}^d$ it holds that $(G_k(t,x))_{k\in \mathbb{Z}}$ are i.i.d. random variables. This and Corollary \ref{cor:1.22.2} (applied for every $t\in [0,T]$, $x\in \mathbb{R}^d$ with $p \curvearrowleft \mathfrak{p}$, $n \curvearrowleft m$, $(X_k)_{k\in \{1,2,\..,m\}} \curvearrowleft (G_k(t,x))_{k\in \{1,2,...,m\}}$), with the notation of Corollary \ref{cor:1.22.2} ensure that for all $t\in [0,T]$, $x \in \mathbb{R}^d$, it holds that: \begin{align} \left( \E \left[ \left| \frac{1}{m} \left[ \sum^{m}_{k=1} G_k(t,x) \right] - \E \left[ G_1(t,x) \right] \right| ^\mathfrak{p} \right] \right)^{\frac{1}{\mathfrak{p}}} \leqslant \frac{\mathfrak{m}}{m^{\frac{1}{2}}}\left(\E \left[|G_1(t,x)|^\mathfrak{p} \right] \right)^{\frac{1}{\mathfrak{p}}} \end{align} \medskip -Combining this, with (1.16), (1.17), and item (ii) of Corollary \ref{cor:1.20.1} yields that: +Combining this, with (\ref{(1.12)}), (\ref{(2.1.4)}), and Item (ii) of Corollary \ref{cor:1.20.1} yields that: \begin{align} &\left(\E\left[\left|U^0(t,x) - \E \left[U^0(t,x)\right]\right|^\mathfrak{p}\right]\right)^\frac{1}{\mathfrak{p}} \nonumber\\ &= \left(\E \left[\left|\frac{1}{m}\left[\sum^{m}_{k=1}G_k(t,x)\right]- \E \left[G_1(t,x)\right]\right|^\mathfrak{p} \right]\right)^{\frac{1}{\mathfrak{p}}} \\ &\leqslant \frac{\mathfrak{m}}{m^{\frac{1}{2}}}\left(\E \left[\left| G_1(t,x)\right| ^\mathfrak{p}\right]\right)^{\frac{1}{\mathfrak{p}}} \\ &= \frac{\mathfrak{m}}{m^{\frac{1}{2}}} \left[\left(\E \left[\left|g\left(x+\mathcal{W}^1_{T-t}\right)\right|^\mathfrak{p}\right]\right)^\frac{1}{\mathfrak{p}}\right] \end{align} -This and the fact that $\mathcal{W}^0$ has independent increments ensure that for all $n\in $, $t\in [0,T]$, $x\in \mathbb{R}^d$ it holds that: +This and the fact that $\mathcal{W}^0$ has independent increments ensure that for all $m\in \N$, $t\in [0,T]$, $x\in \mathbb{R}^d$ it holds that: \begin{align} \left(\E \left[\left| U^0 \left(t,x+\mathcal{W}^0_t\right) - \E \left[U^0 \left(t,x+\mathcal{W}^0_t\right)\right]\right|^\mathfrak{p}\right]\right)^{\frac{1}{\mathfrak{p}}} \leqslant \frac{\mathfrak{m}}{m^{\frac{1}{2}}} \left[\left(\E \left[\left| g \left(x+\mathcal{W}^0_T\right)\right|^\p\right]\right)^{\frac{1}{\mathfrak{p}}} \right] \end{align} @@ -341,7 +341,7 @@ Which in turn yields that: \begin{align}\label{(1.48)} \mathfrak{L}\left( \frac{ \mathfrak{m}}{m^{\frac{1}{2}}}\right)\left(\E \left[ \left( 1+ \left\| x+\mathcal{W}^0_T \right\|_E^p \right)^\p\right]\right)^\frac{1}{\p} \leqslant \mathfrak{L}\left( \frac{ \mathfrak{m}}{m^{\frac{1}{2}}}\right)\left(\sup_{s \in [0,T]}\E \left[ \left( 1+ \left\| x+\mathcal{W}^0_s \right\|_E^p \right)^\p\right]\right)^\frac{1}{\p} \end{align} -Combining \ref{(1.46)}, \ref{(1.47)}, and \ref{(1.48)} yields that: +Combining (\ref{(1.46)}), (\ref{(1.47)}), and (\ref{(1.48)}) yields that: \begin{align} \left( \E \left[ \left| U^0 \left(t,x+\mathcal{W}^0_t \right) - u \left( t, x+\mathcal{W}^0_t \right) \right|^\p \right] \right)^{\frac{1}{\p}} &\leqslant \left( \frac{ \mathfrak{m}}{m^{\frac{1}{2}}}\right)\left(\E \left[\left| g \left(x+\mathcal{W}^0_T\right)\right|^\p\right]\right)^\frac{1}{\p} \nonumber\\ &\les\mathfrak{L}\left( \frac{ \mathfrak{m}}{m^{\frac{1}{2}}}\right)\left(\sup_{s\in[0,T]}\E \left[ \left( 1+ \left\| x+\mathcal{W}^0_s \right\|_E^p \right)^\p\right]\right)^\frac{1}{\p} @@ -379,19 +379,19 @@ Thus we get for all $\mft \in [0,T]$, $x\in \R^d$, $n \in $: This completes the proof of Corollary \ref{cor:1.25.1}. \end{proof} -\begin{theorem}\label{tentpole_1} Let $T,L,p,q, \mathfrak{d} \in [0,\infty), m \in \mathbb{N}$, $\Theta = \bigcup_{n\in \mathbb{N}} \Z^n$, let $g_d\in C(\R^d,\R)$, and assume that $d\in \N$, $t \in [0,T]$, $x = (x_1,x_2,...,x_d)\in \R^d$, $v,w \in \R$ and that $\max \{ |g_d(x)|\} \leqslant Ld^p \left(1+\Sigma^d_{k=1}\left|x_k \right|\right)$, let $\left(\Omega, \mathcal{F}, \mathbb{P}\right)$ be a probability space, let $\mathcal{W}^{d,\theta}: [0,T] \times \Omega \rightarrow \R^d$, $d\in \N$, $\theta \in \Theta$, be independent standard Brownian motions, assume for every $d\in \N$ that $\left(\mathcal{W}^{d,\theta}\right)_{\theta \in \Theta}$ are independent, let $u_d \in C([0,T] \times \R^d,\R)$, $d \in \N$, satisfy for all $d\in \N$, $t\in [0,T]$, $x \in \R^d$ that $\E \left[g_x \left(x+\mathcal{W}^{d,0}_{T-t} \right)\right] < \infty$ and: +\begin{theorem}\label{tentpole_1} Let $T,L,p,q, \mathfrak{d} \in [0,\infty), m \in \mathbb{N}$, $\Theta = \bigcup_{n\in \mathbb{N}} \Z^n$, let $g_d\in C(\R^d,\R)$, and assume that $d\in \N$, $t \in [0,T]$, $x = (x_1,x_2,...,x_d)\in \R^d$, $v,w \in \R$ and that $\max \{ |g_d(x)|\} \leqslant Ld^p \left(1+\Sigma^d_{k=1}\left|x_k \right|^q\right)$, let $\left(\Omega, \mathcal{F}, \mathbb{P}\right)$ be a probability space, let $\mathcal{W}^{d,\theta}: [0,T] \times \Omega \rightarrow \R^d$, $d\in \N$, $\theta \in \Theta$, be independent standard Brownian motions, assume for every $d\in \N$ that $\left(\mathcal{W}^{d,\theta}\right)_{\theta \in \Theta}$ are independent, let $u_d \in C([0,T] \times \R^d,\R)$, $d \in \N$, satisfy for all $d\in \N$, $t\in [0,T]$, $x \in \R^d$ that $\E \left[g_x \left(x+\mathcal{W}^{d,0}_{T-t} \right)\right] < \infty$ and: \begin{align} - u_d\left(t,x\right) = \E \left[g_d \left(x + \mathcal{W}^{d,0}_{T-t}\right)\right] + u_d\left(t,x\right) = \E \left[g_d \left(x + \mathcal{W}^{d}_{T-t}\right)\right] \end{align} Let $U^{d,\theta}_m: [0,T] \times \R^d \times \Omega \rightarrow \R$, $d \in \N$, $m\in \Z$, $\theta \in \Theta$, satisfy for all, $d\in \N$, $m \in \Z$, $\theta \in \Theta$, $t\in [0,T]$, $x\in \R^d$ that: \begin{align} - U^{d,\theta}_m(t,x) = \frac{1}{m} \left[\sum^{m}_{k=1} g_d \left(x + \mathcal{W}^{d,(\theta, 0,-k)}_{T-t}\right)\right] + U^{d}_m(t,x) = \frac{1}{m} \left[\sum^{m}_{k=1} g_d \left(x + \mathcal{W}^{d}_{T-t}\right)\right] \end{align} and for every $d,n,m \in \N$ let $\mathfrak{C}_{d,n,m} \in \Z$ be the number of function evaluations of $u_d(0,\cdot)$ and the number of realizations of scalar random variables which are used to compute one realization of $U^{d,0}_m(T,0): \Omega \rightarrow \R$. There then exists $c \in \R$, and $\mathfrak{N}:\N \times (0,1] \rightarrow \N$ such that for all $d \in \N$, $\varepsilon \in (0,1]$ it holds that: \begin{align}\label{(2.48)} - \sup_{t\in[0,T]} \sup_{x \in [-L,L]^d} \left(\E \left[\left| u_d(t,x) - U^{d,0}_{\mathfrak{N}(d,\epsilon)}\right|^\p\right]\right)^\frac{1}{\p} \leqslant \epsilon + \sup_{t\in[0,T]} \sup_{x \in [-L,L]^d} \left(\E \left[\left| u_d(t,x) - U^{d,0}_{\mathfrak{N}(d,\epsilon)}\right|^\p\right]\right)^\frac{1}{\p} \leqslant \ve \end{align} and: @@ -399,7 +399,7 @@ and: \mathfrak{C}_{d,\mathfrak{N}(d,\varepsilon), \mathfrak{N}(d,\varepsilon)} \leqslant cd^c\varepsilon^{-(2+\delta)} \end{align} \end{theorem} -\begin{proof} Throughout the proof let $\mathfrak{m}_\mathfrak{p} = \sqrt{\mathfrak{p} -1}$, $\mathfrak{p} \in [2,\infty)$, let $\mathbb{F}^d_t \subseteq \mathcal{F}$, $d\in \N$, $t\in [0,T]$ satisfy for all $d \in \N$, $t\in [0,T]$ that: +\begin{proof} Throughout the proof let $\mathfrak{m}_\mathfrak{p} = \fk_p\sqrt{\mathfrak{p} -1}$, $\mathfrak{p} \in [2,\infty)$, let $\mathbb{F}^d_t \subseteq \mathcal{F}$, $d\in \N$, $t\in [0,T]$ satisfy for all $d \in \N$, $t\in [0,T]$ that: \begin{align}\label{2.3.29} \mathbb{F}^d_t = \begin{cases} \bigcap_{s\in[t,T]} \sigma \left(\sigma \left(W^{d,0}_r: r \in [0,s]\right) \cup \{A\in \mathcal{F}: \mathbb{P}(A)=0\}\right) & :t7Z9*e zq_>1BAW{>g6G~`byzl$ob$|C;>;99o*PJtxInO+M&$IWON?bnO_(>(*2d(ESs{C47 z7kvGEoSu4I2%_QXZo2eJ_M+ZUuGkEHLQdj+x2rI5#&`L2>{YE%(Vl_+YT|0YN+CRSX82$nR zausvu?qY;VHQ0|d0`km=#tUj=NpH)F7UR3i%AF37ZmJ#*IZosKQK=)jV~X`7ncCVa zF0+U@lEd1_BN%0U!-=PQ@`4^8eAZjIvFs2%Jy7^@YAMTwNM39xvvkQiqUiU1pRNbw zwNQM0$W3d8-7FS&!bDqoWMrW5Z%LlB-;z!d$tTIeGr{#Bd>X6b6;H7t1Tp{EG9tC* z-b%b!*$U$1i&pT(gN11E>+8!lzOC2vFVCuSuIq>Cd4ZR;69&iF^XdtG#Zd9n2(@_jH^Bu(OA>({A9@E3QR?7)LIQ}!xh=By z9J|S?Mr^-M^DDNj9694z+z1iVGv{Q8k zoI%mgwYnWjY7PlX>VpRe7B$(641w48w$Kj%$zNRg>L$~GIGz^QdIDB4WNi~9FmU=o zMH>rV1PIL~3<<1yMgaRbG7_2x8n156fd>##r)IX8I5yjcBjI94^L~%M=B9MBtPhHU 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