\section[Some Useful Results]{Some Useful Results\protect\footnote{I may omit some technical conditions in this section.}}

\begin{theorem}[Cauchy--Schwarz inequality]
\leavevmode
\begin{enumerate}
    \item (definite integrals) Let $f$ and $g$ be real functions which are continuous on the closed interval $[a, b]$. Then:
    \[
    \left(\int_a^b f(t) g(t) \, dt\right)^2 \leq \int_a^b f^2(t) \, dt \int_a^b g^2(t) \, dt.
    \]
    As a corollary, we have
    \[
    \left(\int_a^b f(t) \, dt\right)^2 \leq (b - a) \int_a^b f^2(t) \, dt.
    \]

    \item (expectations) For any two random variables $X$ and $Y$,
    \[
    [\E(XY)]^2 \leq \E(X^2) \E(Y^2),
    \]
    or equivalently,
    \[
    |\E(XY)| \leq \sqrt{\E(X^2) \E(Y^2)}.
    \]
\end{enumerate}
\end{theorem}

\begin{theorem}[Fubini--Tonelli theorem]
Let $X$ be a stochastic process such that $\int_0^T \E(|X_s|) \, ds < \infty$, then
\[
\E\left[\int_0^T X_s \, ds\right] = \int_0^T \E[X_s] \, ds.
\]
\end{theorem}
