From d4cf8e21422906ca0946a8dfb0d581884ee3a9b4 Mon Sep 17 00:00:00 2001
From: Shakil Rafi <sarafi@uark.edu>
Date: Sat, 24 Feb 2024 15:16:42 -0600
Subject: [PATCH] More stuff added

---
 .DS_Store                                     | Bin 18436 -> 18436 bytes
 .../ann_rep_brownian_motion_monte_carlo.tex   |  22 +++++++++++++++---
 2 files changed, 19 insertions(+), 3 deletions(-)

diff --git a/.DS_Store b/.DS_Store
index 46de23aacc7a3e0fab528a11a4ca6f8ac2e4fb4a..175f0f5dcf7dbb5a121d937642f3a1bad5acf693 100644
GIT binary patch
delta 46
zcmZpfz}PZ@af71n<i7$5;u6)>=B7FdMh0fJIttZ>hNhM}3g(6ull8=mH=owElmY-&
CEDfdr

delta 22
ecmZpfz}PZ@af71nWPi!z$s9`Vn@{UnN&x^`kq3zY

diff --git a/Dissertation/ann_rep_brownian_motion_monte_carlo.tex b/Dissertation/ann_rep_brownian_motion_monte_carlo.tex
index 247eab8..4969022 100644
--- a/Dissertation/ann_rep_brownian_motion_monte_carlo.tex
+++ b/Dissertation/ann_rep_brownian_motion_monte_carlo.tex
@@ -889,7 +889,7 @@ Let $t \in \lp 0,\infty\rp$ and $T \in \lp t,\infty\rp$. Let $\lp \Omega, \mathc
 		&=2\cdot \E\lb \lp \int^t_s d\cW_r^d\rp^2\rb \nonumber\\
 		&=2\cdot \E \lb \lp \cW^d_{t-s}\rp^2\rb 
 	\end{align} 
-\end{proof}
+
 Note now that:
 \begin{align}
 	\va \lb \cW^d_{t-s}\rb &= \E \lb \lp \cW_{t-s}^d\rp^2\rb - \E \lb \cW^d_{t-s}\rb^2 \nonumber \\
@@ -898,11 +898,27 @@ Note now that:
 \end{align}
 Now note that since $\cW^d_r$ are standard Brownian motions, and their expectation and variance are $\mymathbb{0}_d$ and $\mathbb{I}_d$ respectively. Whence it is the case that the probability density function for $\cW_{t-s}^d$ is:
 \begin{align}
-	\lp 2\pi\rp^{-\frac{d}{2}}\lp t-s\rp^{-\frac{1}{2}}\exp \lp \frac{-1}{2(t-s)}\mymathbb{e}_{1,d}\cdot \lb x \rb_*^2\rp
+	\ff_{\cW^d_{t-s}} \lp x\rp= \lp 2\pi\rp^{-\frac{d}{2}}\lp t-s\rp^{-\frac{1}{2}}\exp \lp \frac{-1}{2(t-s)}\mymathbb{e}_{1,d}\cdot \lb x \rb_*^2\rp
 \end{align}
-However $\cX^{d,t,x}_s$ is a shifted normal distribution
+However $\cX^{d,t,x}_s$ is a shifted normal distribution, specifically shifted by $x$. Its p.d.f. is thus:
+\begin{align}
+	\ff_{\cX^{d,t,x}_s}\lp \scrX \rp = \lp 2\pi\rp^{-\frac{d}{2}}\lp t-s\rp^{-\frac{1}{2}}\exp \lp \frac{-1}{2(t-s)}\mymathbb{e}_{1,d}\cdot \lb \scrX +x\rb_*^2\rp
+\end{align}
+The Law of the Unconscious Statistician then says that:
+\begin{align}
+	\E \lb \fu^T_d \lp \cX^{d,t,x}_s\rp\rb = \int_{\R^d}\fu^T_d\lp \scrX\rp\cdot \ff_{\cX^{d,t,x}_s}\lp \scrX\rp d\scrX
+\end{align}
+And further that:
+\begin{align}
+	\E \lb \alpha_d \lp \cX^{d,t,x}_s\rp\rb = \int_{\R^d} \alpha_d\lp \scrX\rp\cdot \ff_{\cX^{d,t,x}_s}\lp \scrX\rp d\scrX
+\end{align}
+
+\textcolor{red}{\textbf{Need to re-examine $\fu^T_d, \alpha_d$}}
+
+Now note this that
 
 
+\end{proof}