Making progress towards the last proof. I might just give up and do Monte Carlo Standard Error and be done with it

Dissertation/ann_rep_brownian_motion_monte_carlo.tex

@@ -601,7 +601,6 @@ Let $t \in \lp 0,\infty\rp$ and $T \in \lp t,\infty\rp$. Let $\lp \Omega, \mathc
 		\mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i} = \mathsf{UE}^{N,h,q,\ve}_{n, \mathsf{G}_d} \bullet \lb \tun^{N+1}_1 \boxminus \aff_{\mymathbb{0}_{d,d},\mathcal{X}_{\omega_i}} \rb 
 	\end{align}
 	It is then the case that for all $\fx = \{x_0,x_1,\hdots, x_N\} \in \R^{N+1}$ and $x \in \R^d$ that:
-		\item It is also the case that:
 		\begin{align}
 		&\left| \exp \lp \int^T_t fds\rp \mathfrak{u}_d^T\lp x\rp - \real_{\rect}\lp \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rp \lp f\lp \lb \fx\rb_*\rp \frown x\rp \right|\nonumber\\ &\les  3\ve +2\ve \left| \mathfrak{u}_d^T\lp t,x\rp\right|^q+2\ve \left| \exp \lp \int^b_afdx\rp\right|^q + \ve \left| \exp \lp \int^b_afdx\rp - \mathfrak{e}\right|^q -\mathfrak{e}\mathfrak{u}_d^T\lp x \rp\nonumber
 		\end{align}
@@ -914,8 +913,15 @@ And further that:
 \end{align}
 
 \textcolor{red}{\textbf{Need to re-examine $\fu^T_d, \alpha_d$}}
+Note that It\^o's Lemma allows us to conclude that:
+\begin{align}
+	d\:\alpha_d \lp \cX^{d,t,x}_s\rp = \alpha_d^{'}\lp \cX^{d,t,x}_s\rp d\cX_t+\alpha_d^{''}\lp \cX_t\rp dt
+\end{align}
 
-Now note this that
+Now note this that Fubini's theorem states that:
+\begin{align}\label{fubinis_to_integral}
+	\E \lb \int^T_t \alpha_d \circ \cX^{d,t,x}_s ds\rb = \int^T_t \E \lb \alpha_d\circ \cX^{d,t,x}_s\rb ds
+\end{align}
 
 
 \end{proof}