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Shakil Rafi 2024-03-01 18:10:11 -06:00
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Dissertation/ann_rep_brownian_motion_monte_carlo.tex
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@ -841,7 +841,7 @@ Let $t \in \lp 0,\infty\rp$ and $T \in \lp t,\infty\rp$. Let $\lp \Omega, \mathc
Observe that the absolute homogeneity condition for norms, the fact that the Brownian motions are independent of each other, Lemma \ref{lem:sm_sum}, the fact that $\mathfrak{n}\in \N$, the fact that the upper limit of error remains bounded by the same bound for all $\omega_i \in \Omega$, and Lemma \ref{sum_of_errors_of_stacking}, then yields us: Observe that the absolute homogeneity condition for norms, the fact that the Brownian motions are independent of each other, Lemma \ref{lem:sm_sum}, the fact that $\mathfrak{n}\in \N$, the fact that the upper limit of error remains bounded by the same bound for all $\omega_i \in \Omega$, and Lemma \ref{sum_of_errors_of_stacking}, then yields us:
\begin{align} \begin{align}
&\left| \frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1}\lb \exp \lp \int_t^T f\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp ds \cdot u_d^T\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rp\rb \rb - \real_{\rect}\lp \mathsf{UES}^{N,h,q,\ve}_{n,\mathsf{G}_d,\Omega, \fn}\rp\right|\nonumber \\ &\left| \frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1}\lb \exp \lp \int_t^T f\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp ds \cdot u_d^T\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rp\rb \rb - \real_{\rect}\lp \mathsf{UES}^{N,h,q,\ve}_{n,\mathsf{G}_d,\Omega, \fn}\rp\right|\nonumber \\
&\left| \frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1}\lb \exp \lp \int_t^T f\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp ds \cdot u_d^T\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rp\rb \rb - \real_{\rect}\lb \frac{1}{\mathfrak{n}} \triangleright\lp \sm_{\mathfrak{n},1}\bullet\lb \boxminus_{i=1}^{\mathfrak{n}} \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rb\rp\rb\right| \nonumber \\ &=\left| \frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1}\lb \exp \lp \int_t^T f\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp ds \cdot u_d^T\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rp\rb \rb - \real_{\rect}\lb \frac{1}{\mathfrak{n}} \triangleright\lp \sm_{\mathfrak{n},1}\bullet\lb \boxminus_{i=1}^{\mathfrak{n}} \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rb\rp\rb\right| \nonumber \\
&\les \left|\frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1} \exp \lp \int_t^T f\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp ds \cdot u_d^T\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rp \rb - \frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1}\lp \real_{\rect}\lb \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rb\rp \rb \right| \nonumber \\ &\les \left|\frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1} \exp \lp \int_t^T f\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp ds \cdot u_d^T\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rp \rb - \frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1}\lp \real_{\rect}\lb \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rb\rp \rb \right| \nonumber \\
&\les \cancel{\frac{1}{\mathfrak{n}} \sum^{\mathfrak{n}}_{i=1}}\left| \exp \lp \int^T_tf\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp ds \cdot u^T_d\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rp - \real_{\rect}\lp \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i} \rp \right| \nonumber\\ &\les \cancel{\frac{1}{\mathfrak{n}} \sum^{\mathfrak{n}}_{i=1}}\left| \exp \lp \int^T_tf\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp ds \cdot u^T_d\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rp - \real_{\rect}\lp \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i} \rp \right| \nonumber\\
&\les \left| \exp \lp \int^T_tf\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp ds \cdot u^T_d\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rp - \real_{\rect}\lp \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rp \right| \nonumber \\ &\les \left| \exp \lp \int^T_tf\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp ds \cdot u^T_d\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rp - \real_{\rect}\lp \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rp \right| \nonumber \\
@ -866,7 +866,7 @@ Let $t \in \lp 0,\infty\rp$ and $T \in \lp t,\infty\rp$. Let $\lp \Omega, \mathc
\begin{corollary} \begin{corollary}
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: 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} \begin{align}
\left\| \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-\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 u_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 u\lp T,\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 u_d^T\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rb \rb \right|\nonumber
\end{align} \end{align}
\end{corollary} \end{corollary}
\begin{proof} \begin{proof}