Trying out different things for finding the expectation
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@ -1116,7 +1116,7 @@ Let $\mathfrak{p}_i$ for $i \in \{1,2,...\}$ be the set of functions defined for
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\wid_{\hid \lp \lb \pwr^{q,\ve}_{n-1} \boxminus\tun_{\dep \lp \pwr_{n-1}^{q,\ve} \rp}\rb\rp} \lp \lb \pwr^{q,\ve}_{n-1} \boxminus\tun_{\dep \lp \pwr_{n-1}^{q,\ve} \rp}\rb\rp = 24+2=26
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\end{align}
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Finally Lemma \ref{comp_prop}, (\ref{(6.2.34)}), and Corollary \ref{cor:sameparal}, also tells us that:
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Finally Lemma \ref{comp_prop}, (\ref{(6.2.34)}), a geometric series argument, and Corollary \ref{cor:sameparal}, also tells us that:
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\begin{align}
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&\param \lp \pwr_{n}^{q,\ve}\rp\\ &= \param \lp \prd^{q,\ve} \bullet\lb \pwr^{q,\ve}_{n-1} \boxminus\tun_{\dep \lp \pwr_{n-1}^{q,\ve} \rp}\rb \bullet \cpy_{2,1}\rp \nonumber \\
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&= \param \lp \prd^{q,\ve} \bullet \lb \pwr^{q,\ve}_{n-1} \boxminus\tun_{\dep \lp \pwr_{n-1}^{q,\ve} \rp}\rb\rp \nonumber \\
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@ -534,7 +534,8 @@ Let $n, N,h\in \N$. Let $\delta,\ve \in \lp 0,\infty \rp $, $q\in \lp 2,\infty \
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\end{align}
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\item It is also the case that:
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\begin{align}
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&\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
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&\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\\
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&\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
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\end{align}
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Where, as per Lemma \ref{mathsfE}, $\mathfrak{e}$ is defined as:
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\begin{align}
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@ -695,12 +696,7 @@ Note that for a fixed $T \in \lp 0,\infty \rp$ it is the case that $u_d\lp t,x \
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\end{center}
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\end{remark}
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\section{The $\mathsf{UES}^{N,h,q,\ve}_{n,\mathsf{G}_d,\Omega,\fn}$ network}
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\begin{definition}[The Kahane-Kintchine Constant]
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Let $p,q \in \lp 0,\infty\rp$. We will then denote by $\fK_{p,q}\in \lb 0,\infty\rb$, the extended real number given by:
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\begin{align}
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\fK_{p,q} = \sup \left\{ c \in \lb 0,\infty \rp : \lb \exists \text{ an }\R-\text{Banach Space} \rb \right\}
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\end{align}
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\end{definition}
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\begin{lemma}\label{lem:sm_sum}
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Let $\nu_1,\nu_2,\hdots, \nu_n \in \neu$ such that for all $i \in \{1,2,\hdots, n\}$ it is the cast that $\out\lp \nu_i\rp = 1$, and it is also the case that $\dep \lp \nu_1 \rp = \dep \lp \nu_2 \rp = \cdots =\dep \lp \nu_n\rp$. Let $x_1 \in \R^{\inn\lp \nu_1\rp},x_2 \in \R^{\inn\lp \nu_2\rp},\hdots, x_n \in \R^{\inn\lp \nu_n\rp}$ and $\fx \in \R^{\sum_{i=1}^n \inn \lp \nu_i\rp}$. It is then the case that we have that:
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\begin{align}
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@ -743,10 +739,21 @@ Note that for a fixed $T \in \lp 0,\infty \rp$ it is the case that $u_d\lp t,x \
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\end{align}
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This proves the inductive case and hence the Lemma.
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\end{proof}
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\begin{lemma}
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Let, $\lp \Omega, \mathcal{F}, \mathbb{P}\rp$ be a probability space and let $\mathcal{X}: \Omega \rightarrow \R^d$ be a random variable with $\E\lb\mathcal{X}\rb = \mu < \infty$, and probability density function $\ff_{\cX }$. Let $g: \R^d \rightarrow \R$ be a measurable continuous function. It is then the case that
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\begin{lemma}\label{var_of_rand}
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Let $\lp \Omega, \cF, \mathbb{P} \rp$ be a probability space. Let $X_d: \Omega \rightarrow \R_d$ be a random variable. Let $f: \R_d \rightarrow \R$ be a function such that for all $x,\fx \in \R^d$ it is the case that $\left\| f\lp x\rp - f\lp \fx\rp\right\|_E \les \fL\left| x-\fx\right|$. It is then the case that $\var\lb f\lp X_d\rp\rb \les 2\fL^2\var\lb X_d\rb$.
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\end{lemma}
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\begin{proof}
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Let $\fX_d$ be an i.i.d. copy of $X_d$. As such it is the case that $\cov \lp X_d, \fX_d\rp = 0$, whence it is the case that $\var\lb X_d, \fX_d\rb = \var\lb X_d\rb + \var\lb \fX_d\rb = \var[X_d] + \var\lb -\fX_d\rb = \var\lb X_d - \fX_d\rb = 2\var\lb X_d\rb$. Note that $f\lp X_d\rp$ and $f\lp \fX_d\rp$ are also indepentend and thus $\cov\lp f\lp X_d\rp,f\lp \fX_d\rp\rp = 0$, and whence we get that $\var\lb f\lp X_d\rp - f\lp \fX_d\rp\rb = 2\var \lb \fX_d\rb$. This then yields that:
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\begin{align}
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2\var \lb f\lp X_d\rp\rb &= \var\lb f\lp X_d\rp - f\lp \fX_d\rp\rb \nonumber\\
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&= \E \lb \lp f\lp X_d\rp -f\lp \fX_d\rp\rp^2\rb - \lp \E \lb f\lp X_d\rp - f\lp \fX_d\rp\rb\rp^2 \nonumber \\
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&= \E \lb \lp f\lp X_d\rp -f\lp \fX_d\rp\rp^2\rb \nonumber\\
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&= \fL^2\cdot \E \lb \lp X_d - \fX_d \rp^2\rb \nonumber\\
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&= \fL^2\cdot 2 \var \lb X_d\rb \nonumber\\
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\implies \var \lb f\lp X_d\rp\rb &= \fL^2\cdot \var\lb X_d\rb
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\end{align}
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This proves the Lemma.
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\end{proof}
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\begin{lemma}[R\textemdash, 2024, Approximants for Brownian Motion]
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Let $t \in \lp 0,\infty\rp$ and $T \in \lp t,\infty\rp$. Let $\lp \Omega, \mathcal{F}, \mathbb{P}\rp$ be a probability space. Let $n,N\in \N$, and $h \in \lp 0, \infty \rp$. Let $\delta,\ve \in \lp 0,\infty \rp $, $q\in \lp 2,\infty \rp$, satisfy that $\delta = \ve \lp 2^{q-1} +1\rp^{-1}$. Let $f:[t, T] \rightarrow \R$ be continuous almost everywhere in $\lb t, T \rb$. Let it also be the case that $f = g \circ \fh$, where $\fh: \lb t,T\rb \rightarrow \R^d$, and $g: \R^d \rightarrow \R$. Let $t=t_0 \les t_1\les \cdots \les t_{N-1} \les t_N=T$ such that for all $i \in \{0,1,...,N\}$ it is the case that $h = \frac{T-t}{N}$, and $t_i = t_0+i\cdot h$ . Let $\mathbf{t} = \lb t_0 \: t_1\: \cdots t_N \rb$ and as such let $f\lp\lb \mathbf{t} \rb_{*,*} \rp = \lb f(t_0) \: f(t_1)\: \cdots \: f(t_N) \rb$. Let $u_d \in C \lp \R^d,\R\rp$ satisfy for all $d \in \N$, $t \in \lb 0,T\rb$, $x \in \R^d$ that:
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@ -795,7 +802,7 @@ Let $t \in \lp 0,\infty\rp$ and $T \in \lp t,\infty\rp$. Let $\lp \Omega, \mathc
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\end{align}
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\item It is also the case that:
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\begin{align}
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&\left| \mathbb{E} \lb \exp \lp \int^T_t f\lp \mathcal{X}^{d,t,x}_{r}\rp ds\rp u_d\lp T,\mathcal{X}^{d,t,x}_{r,\omega_i}\rp \rb - \real_{\rect}\lp \mathsf{UES}^{N,h,q,\ve}_{n,\mathsf{G}_d,\Omega,\fn}\rp\right| \\
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&\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\\
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&\les 3\ve +2\ve \left| \fu^T_d\lp 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}\fu^T_d\lp x \rp\nonumber
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\end{align}
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Where, as per Lemma \ref{mathsfE}, $\mathfrak{e}$ is defined as:
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@ -831,100 +838,177 @@ Let $t \in \lp 0,\infty\rp$ and $T \in \lp t,\infty\rp$. Let $\lp \Omega, \mathc
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&\les \mathfrak{n}^2\cdot \param \lp \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rp \nonumber \\
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&\les \fn^2 \cdot \lb \frac{360q}{q-2} \lb \log_2 \lp \ve^{-1} \rp +q+1 \rb +324+ 48n\right. \nonumber\\ &\left. +24 \wid_{\hid\lp \mathsf{G}_d\rp}\lp \mathsf{G}_d\rp + 4\max \left\{\param \lp \mathsf{E}^{N,h,q,\ve}_{n}\rp, \param \lp \mathsf{G}_d\rp \right\} \rb
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\end{align}
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Now observe that by the triangle inequality, we have that:
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\begin{align}
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&\left| \E \lb \exp \lp \int^T_t f ds\rp \fu_d^T\lp x\rp\rb - \real_{\rect}\lp \mathsf{UES}^{N,h,q,\ve}_{n,\mathsf{G}_d,\Omega, \fn}\rp \right| \label{big_eqn_lhs} \\
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&=\left| \E \lb \exp \lp \int^T_t f ds\rp \fu_d^T\lp x\rp \rb - \inst_{\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\\
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&\les \left| \E \lb \exp \lp \int^T_t f ds\rp \fu_d^T\lp x\rp \rb - \frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1}\lb \exp \lp \int_t^T f ds \rp \cdot \fu_d^T\lp x\rp\rb \rb \right|\label{big_eqn_rhs_summand_1} \\
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&+\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 \rp \cdot \fu_d^T\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\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| \label{big_eqn_lhs_summand_2}
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\end{align}
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Observe that by the triangle inequality, 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 renders the second summand, (\ref{big_eqn_lhs_summand_2}), as:
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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:
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\begin{align}
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&\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 \\
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&\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 \\
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&\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 \\
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&\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\\
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&\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
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&\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 \\
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&\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
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\end{align}
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This renders (\ref{big_eqn_lhs}) as:
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\end{proof}
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% Now observe that by the triangle inequality, we have that:
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% \begin{align}
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% &\left| \E \lb \exp \lp \int^T_t f ds\rp \fu_d^T\lp x\rp\rb - \real_{\rect}\lp \mathsf{UES}^{N,h,q,\ve}_{n,\mathsf{G}_d,\Omega, \fn}\rp \right| \label{big_eqn_lhs} \\
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% &=\left| \E \lb \exp \lp \int^T_t f ds\rp \fu_d^T\lp x\rp \rb - \inst_{\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\\
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% &\les \left| \E \lb \exp \lp \int^T_t f ds\rp \fu_d^T\lp x\rp \rb - \frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1}\lb \exp \lp \int_t^T f ds \rp \cdot \fu_d^T\lp x\rp\rb \rb \right|\label{big_eqn_rhs_summand_1} \\
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% &+\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 \rp \cdot \fu_d^T\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\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| \label{big_eqn_lhs_summand_2}
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% \end{align}
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% Observe that by the triangle inequality, 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 renders the second summand, (\ref{big_eqn_lhs_summand_2}), as:
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% \begin{align}
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% &\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 \\
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% &\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 \\
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% &\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\\
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% &\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
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% \end{align}
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\begin{corollary}
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Let $N,n,\fn \in \N$, $h,\ve \in \lp 0,\infty\rp$, $q\in\lp 2,\infty\rp$, given $\mathsf{UES}^{N,h,q,\ve}_{n,\mathsf{G}_d, \Omega, \fn}$, the Monte Carlo standard error for approximating $\exp \lp \int_t^T f\lp \mathcal{X}^{d,t,x}_{r,\Omega}\rp ds \cdot u_d^T\lp \mathcal{X}^{d,t,x}_{r,\Omega}\rp\rp$ is:
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\begin{align}
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&\left| \E \lb \exp \lp \int^T_t f ds\rp \fu_d^T\lp x\rp\rb - \real_{\rect}\lp \mathsf{UES}^{N,h,q,\ve}_{n,\mathsf{G}_d,\Omega, \fn}\rp \right| \nonumber \\
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&\les \left| \E \lb \exp \lp \int^T_t f ds\rp \fu_d^T\lp x\rp \rb - \frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1}\lb \exp \lp \int_t^T f ds \rp \cdot \fu_d^T\lp x\rp\rb \rb \right| \nonumber \\
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&+\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|
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s
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\end{align}
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Taking the expectation on both sides of this inequality, and applying the linearity and monotonicity of expectation yields:
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\end{corollary}
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\begin{proof}
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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:
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\begin{align}
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&\E \lb \left| \E \lb \exp \lp \int^T_t f ds\rp \fu_d^T\lp x\rp\rb - \real_{\rect}\lp \mathsf{UES}^{N,h,q,\ve}_{n,\mathsf{G}_d,\Omega, \fn}\rp \right|\rb \label{big_eqn_stage_2_lhs}\\
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&\les \E \lb \left| \E \lb \exp \lp \int^T_t f ds\rp \fu_d^T\lp x\rp \rb - \frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1}\lb \exp \lp \int_t^T f ds \rp \cdot \fu_d^T\lp x\rp\rb \rb \right|\rb \label{big_eqn_stage_2_rhs_1} \\
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&+\E\lb \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|\rb \label{big_eqn_stage_2_rhs_2}
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\var \lb \fu^T\lp x\rp\rb &= \lb \nabla \fu^T \lp x\rp\rb^\intercal \cdot \mathbb{I}_d \cdot \nabla \fu^T\lp x\rp + \frac{1}{2}\cdot \Trace\lp \Hess_x^2 \lp f\rp\lp x\rp\rp \nonumber \\
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&= \lb \nabla \fu^T\lp x\rp \rb_*^2 + \frac{1}{2}\cdot \Trace\lp \Hess_x^2\lp f\rp\lp x\rp\rp
|
||||
\end{align}
|
||||
Consider now, the Lyapunov inequality applied to (\ref{big_eqn_stage_2_rhs_1}), which renders it as:
|
||||
We will call the right hand side of the equation above as $\fU.$
|
||||
|
||||
For the second factor in our product consider the following:
|
||||
\begin{align}
|
||||
&\E \lb \left| \E \lb \exp \lp \int^T_t f ds\rp \fu_d^T\lp x\rp \rb - \frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1}\lb \exp \lp \int_t^T f ds \rp \cdot \fu_d^T\lp x\rp\rb \rb \right|\rb \nonumber\\
|
||||
&\les \lp \E \lb \left| \E \lb \exp \lp \int^T_t f ds\rp \fu_d^T\lp x\rp \rb - \frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1}\lb \exp \lp \int_t^T f ds \rp \cdot \fu_d^T\lp x\rp\rb \rb \right|^2\rb \rp^{\frac{1}{2}} \label{where_grohs_will be applied}
|
||||
\cY^{d,t}_{x,s} = \int_t^T\alpha_d \circ \cX^{d,t,x}_{r,\Omega}ds
|
||||
\end{align}
|
||||
Then, \cite[Corollary~2.6]{grohsetal} applied to (\ref{where_grohs_will be applied}), then yields that:
|
||||
Whose Reimann sum, with $\Delta t = \frac{T-t}{n}$ and $t_k = t+k\Delta t$, and Lemma \ref{var_of_rand} is thus rendered as:
|
||||
\begin{align}
|
||||
&\lp \E \lb \left| \E \lb \exp \lp \int^T_t f ds\rp \fu_d^T\lp x\rp \rb - \frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1}\lb \exp \lp \int_t^T f ds \rp \cdot \fu_d^T\lp x\rp\rb \rb \right|^2\rb \rp^{\frac{1}{2}} \nonumber\\
|
||||
&\les 2\sqrt{\frac{1}{\fn}} \lp \E \lb \left| \E \lb \exp \lp \int^T_t f ds\rp \fu_d^T\lp x\rp \rb \right|^2\rb \rp^{\frac{1}{2}}
|
||||
\cY_n &= \Delta t \lb \sum^{n-1}_{k=0} \alpha \circ \cX^{d,t,x}_{r,\Omega}\lp t_k\rp\rb \nonumber\\
|
||||
\var\lb \cY_n \rb &= \var \lb \Delta_t\sum^{n-1}_{k=0}\alpha \circ \cX^{d,t,x}_{r,\Omega}\lp t_k\rp\rb \nonumber\\
|
||||
&= \lp\Delta t\rp^2 \sum^{n-1}_{k=0}\lb \var \lb \alpha \circ \cX^{d,t,x}_{r,\Omega}\lp t_k\rp \rb\rb \nonumber\\
|
||||
&\les \lp \Delta t\rp^2 \sum^{n-1}_{k=0}\lb \fL^2\cdot \var\lp \cX^{d,t,x}_{r,\Omega}\lp t_k\rp\rp\rb \nonumber\\
|
||||
&=\lp \fL\Delta t\rp^2 \sum^{n-1}_{k=0}\lb \var \lp \cX^{d,t,x}_{r,\Omega}\lp t_k\rp\rp \rb
|
||||
\end{align}
|
||||
Looking back at (\ref{big_eqn_stage_2_rhs_2}), we see that the monotonicity and linearity of expectation tells us that:
|
||||
\textbf{Alternatively}:
|
||||
\begin{align}
|
||||
&\E\lb \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|\rb \\
|
||||
&\les \E \lb 3\ve +2\ve \left| \fu^T_d\lp 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}\fu^T_d\lp x \rp\rb \\
|
||||
&\les 3\ve +2\ve \cdot\E\lb \left| \fu^T_d\lp x\rp\right|^q\rb + 2\ve \cdot \E \lb \left| \exp \lp \int^b_afdx\rp\right|^q\rb + \ve \cdot\E \lb \left| \exp \lp \int^b_a f dx\rp - \mathfrak{e}\right|^q\rb -\fe\cdot \E \lb \fu_d^T \lp x\rp\rb \nonumber\\
|
||||
\end{align}
|
||||
Note that:
|
||||
\begin{align}
|
||||
\E\lb \mathcal{X}^{d,t,x}_s\rb &= \E\lb x + \int^t_s \sqrt{2} d\mathcal{W}^d_r\rb \nonumber\\
|
||||
&\les x + \sqrt{2}\cdot\E \lb \int^t_s d\mathcal{W}^d_r \rb \\
|
||||
&= x + \sqrt{2}\cdot \E \lb \mathcal{W}^d_{t-s}\rb \\
|
||||
&= x
|
||||
\end{align}
|
||||
Consider now:
|
||||
\begin{align}
|
||||
\va \lb \cX^{d,t,x}_s\rb &= \va \lb x + \int^t_s \sqrt{2}d\cW^d_r\rb \nonumber \\
|
||||
&= \E \lb\lp x+\int^t_s\sqrt{2}d\cW^d_r - \E \lb x+\int^t_s\sqrt{2}d\cW^d_r\rb\rp^2\rb \nonumber\\
|
||||
&=\E \lb\lp x+\int^t_s\sqrt{2}d\cW^d_r -x\rp^2\rb \nonumber \\
|
||||
&=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
|
||||
&\var \lb \int_t^T\alpha \circ \cX\rb \\
|
||||
&=\E \lb \lp \int^T_t \alpha \circ \cX \rp^2\rb - \lp \E \lb \int^T_t \alpha \circ \cX \rb\rp^2 \\
|
||||
&=\E \lb \int^T_t\lp \alpha \circ \cX \rp^2\rb - \lp \int_t^T \E \lb \alpha \circ \cX \rb\rp^2 \\
|
||||
&=
|
||||
\end{align}
|
||||
|
||||
Note now that:
|
||||
Note that since $\alpha_d$ is Lipschitz with constant $\fL$ we may say that for $\fX^x_t = \cX_t -x$ 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 \\
|
||||
\E\lb \lp \cW^d_{t-s}\rp^2\rb &= \lp t-s \rp\mathbb{I}_d \\
|
||||
2\cdot \E\lb \lp \cW^d_{t-s}\rp^2\rb &= 2\lp t-s\rp\mathbb{I}_d
|
||||
\left| \alpha_d\circ \fX^x_t -\alpha_d \circ \fX^x_0 \right| &\les \fL \cdot\left|\fX^x_t - \fX^x_0\right| \nonumber\\
|
||||
\implies \left| \alpha_d \circ \fX^x_t - \alpha_d\lp 0\rp\right| &\les \fL \left| \fX^x_t-0\right| \nonumber \\
|
||||
\implies \alpha_d \circ \fX^x_t &\les \alpha_d\lp 0\rp + \fL t
|
||||
\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:
|
||||
Thus it is the case that:
|
||||
\begin{align}
|
||||
\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, 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
|
||||
\left| \E \lb \int^T_t \alpha_d \circ \fX_s^t ds \rb\right| &\les \left| \E \lb \int^T_t \alpha_d \lp 0\rp + \fL s ds\rb\right| \nonumber\\
|
||||
&\les \left| \E \lb \int^T_t\alpha_d\lp 0\rp ds +\int^T_t \fL s ds\rb\right| \nonumber\\
|
||||
&\les |\alpha_d\lp 0\rp |\lp T-t\rp + \fL \lp \frac{T^2-t^2}{2} \rp
|
||||
\end{align}
|
||||
We will call the right hand side as $\mathfrak{E}$.
|
||||
|
||||
\textcolor{red}{\textbf{Need to re-examine $\fu^T_d, \alpha_d$}}
|
||||
Note that It\^o's Lemma allows us to conclude that:
|
||||
And it is also the case 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
|
||||
\left| \E \lb \lp \int^T_t \alpha_d \circ \fX^x_t \rp^2\rb\right| &\les \left| \E \lb \iint_{s,\fs=t}^T \lp \alpha_d \circ \fX^x_s\rp\lp \alpha_d \circ \fX^x_\fs\rp\rb dsd\fs\right| \nonumber\\
|
||||
&\les |\alpha_d\lp 0\rp|^2\lp T-t\rp^2 +2\fL |\alpha_d\lp 0\rp |\lp T-t\rp\lp \frac{T^2-t^2}{2}\rp + \fL^2\lp \frac{T^2-t^2}{2}\rp \nonumber
|
||||
\end{align}
|
||||
|
||||
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
|
||||
Thus it is the case that:
|
||||
\begin{align}
|
||||
\var\lp \int_t^T\alpha_d \circ \fX^x_t\rp &\les |\alpha_d\lp 0\rp|^2\lp T-t\rp^2 +2\fL |\alpha_d\lp 0\rp |\lp T-t\rp\lp \frac{T^2-t^2}{2}\rp + \fL^2\lp \frac{T^2-t^2}{2}\rp \nonumber\\
|
||||
&+ |\alpha_d\lp 0\rp |\lp T-t\rp + \fL \lp \frac{T^2-t^2}{2} \rp \nonumber
|
||||
\end{align}
|
||||
Denote the right hand side of the equation above as $\fV$. The the variance vecomes:
|
||||
|
||||
|
||||
\end{proof}
|
||||
\begin{corollary}
|
||||
We may see that
|
||||
\end{corollary}
|
||||
|
||||
% This renders (\ref{big_eqn_lhs}) as:
|
||||
% \begin{align}
|
||||
% &\left| \E \lb \exp \lp \int^T_t f ds\rp \fu_d^T\lp x\rp\rb - \real_{\rect}\lp \mathsf{UES}^{N,h,q,\ve}_{n,\mathsf{G}_d,\Omega, \fn}\rp \right| \nonumber \\
|
||||
% &\les \left| \E \lb \exp \lp \int^T_t f ds\rp \fu_d^T\lp x\rp \rb - \frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1}\lb \exp \lp \int_t^T f ds \rp \cdot \fu_d^T\lp x\rp\rb \rb \right| \nonumber \\
|
||||
% &+\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|
|
||||
% \end{align}
|
||||
% Taking the expectation on both sides of this inequality, and applying the linearity and monotonicity of expectation yields:
|
||||
% \begin{align}
|
||||
% &\E \lb \left| \E \lb \exp \lp \int^T_t f ds\rp \fu_d^T\lp x\rp\rb - \real_{\rect}\lp \mathsf{UES}^{N,h,q,\ve}_{n,\mathsf{G}_d,\Omega, \fn}\rp \right|\rb \label{big_eqn_stage_2_lhs}\\
|
||||
% &\les \E \lb \left| \E \lb \exp \lp \int^T_t f ds\rp \fu_d^T\lp x\rp \rb - \frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1}\lb \exp \lp \int_t^T f ds \rp \cdot \fu_d^T\lp x\rp\rb \rb \right|\rb \label{big_eqn_stage_2_rhs_1} \\
|
||||
% &+\E\lb \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|\rb \label{big_eqn_stage_2_rhs_2}
|
||||
% \end{align}
|
||||
% Consider now, the Lyapunov inequality applied to (\ref{big_eqn_stage_2_rhs_1}), which renders it as:
|
||||
% \begin{align}
|
||||
% &\E \lb \left| \E \lb \exp \lp \int^T_t f ds\rp \fu_d^T\lp x\rp \rb - \frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1}\lb \exp \lp \int_t^T f ds \rp \cdot \fu_d^T\lp x\rp\rb \rb \right|\rb \nonumber\\
|
||||
% &\les \lp \E \lb \left| \E \lb \exp \lp \int^T_t f ds\rp \fu_d^T\lp x\rp \rb - \frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1}\lb \exp \lp \int_t^T f ds \rp \cdot \fu_d^T\lp x\rp\rb \rb \right|^2\rb \rp^{\frac{1}{2}} \label{where_grohs_will be applied}
|
||||
% \end{align}
|
||||
% Then, \cite[Corollary~2.6]{grohsetal} applied to (\ref{where_grohs_will be applied}), then yields that:
|
||||
% \begin{align}
|
||||
% &\lp \E \lb \left| \E \lb \exp \lp \int^T_t f ds\rp \fu_d^T\lp x\rp \rb - \frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1}\lb \exp \lp \int_t^T f ds \rp \cdot \fu_d^T\lp x\rp\rb \rb \right|^2\rb \rp^{\frac{1}{2}} \nonumber\\
|
||||
% &\les 2\sqrt{\frac{1}{\fn}} \lp \E \lb \left| \E \lb \exp \lp \int^T_t f ds\rp \fu_d^T\lp x\rp \rb \right|^2\rb \rp^{\frac{1}{2}}
|
||||
% \end{align}
|
||||
% Looking back at (\ref{big_eqn_stage_2_rhs_2}), we see that the monotonicity and linearity of expectation tells us that:
|
||||
% \begin{align}
|
||||
% &\E\lb \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|\rb \\
|
||||
% &\les \E \lb 3\ve +2\ve \left| \fu^T_d\lp 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}\fu^T_d\lp x \rp\rb \\
|
||||
% &\les 3\ve +2\ve \cdot\E\lb \left| \fu^T_d\lp x\rp\right|^q\rb + 2\ve \cdot \E \lb \left| \exp \lp \int^b_afdx\rp\right|^q\rb + \ve \cdot\E \lb \left| \exp \lp \int^b_a f dx\rp - \mathfrak{e}\right|^q\rb -\fe\cdot \E \lb \fu_d^T \lp x\rp\rb \nonumber\\
|
||||
% \end{align}
|
||||
% Note that:
|
||||
% \begin{align}
|
||||
% \E\lb \mathcal{X}^{d,t,x}_s\rb &= \E\lb x + \int^t_s \sqrt{2} d\mathcal{W}^d_r\rb \nonumber\\
|
||||
% &\les x + \sqrt{2}\cdot\E \lb \int^t_s d\mathcal{W}^d_r \rb \\
|
||||
% &= x + \sqrt{2}\cdot \E \lb \mathcal{W}^d_{t-s}\rb \\
|
||||
% &= x
|
||||
% \end{align}
|
||||
% Consider now:
|
||||
% \begin{align}
|
||||
% \va \lb \cX^{d,t,x}_s\rb &= \va \lb x + \int^t_s \sqrt{2}d\cW^d_r\rb \nonumber \\
|
||||
% &= \E \lb\lp x+\int^t_s\sqrt{2}d\cW^d_r - \E \lb x+\int^t_s\sqrt{2}d\cW^d_r\rb\rp^2\rb \nonumber\\
|
||||
% &=\E \lb\lp x+\int^t_s\sqrt{2}d\cW^d_r -x\rp^2\rb \nonumber \\
|
||||
% &=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}
|
||||
%
|
||||
%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 \\
|
||||
% \E\lb \lp \cW^d_{t-s}\rp^2\rb &= \lp t-s \rp\mathbb{I}_d \\
|
||||
% 2\cdot \E\lb \lp \cW^d_{t-s}\rp^2\rb &= 2\lp t-s\rp\mathbb{I}_d
|
||||
%\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}
|
||||
% \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, 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 $\fu^T_d, \alpha_d$}}
|
||||
%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 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}
|
||||
|
||||
|
||||
|
||||
|
|
|
|||
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|
|
@ -88,7 +88,7 @@
|
|||
\DeclareMathOperator{\supp}{supp}
|
||||
\DeclareMathOperator{\domain}{Domain}
|
||||
\DeclareMathOperator{\lip}{Lip}
|
||||
\DeclareMathOperator{\var}{Var}
|
||||
\DeclareMathOperator{\var}{\mathbb{V}}
|
||||
\DeclareMathOperator{\cov}{Cov}
|
||||
\DeclareMathOperator{\unif}{Unif}
|
||||
\DeclareMathOperator{\dropout}{Dropout}
|
||||
|
|
|
|||
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Loading…
Reference in New Issue