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@ -183,21 +183,22 @@ Typically $\cF_n = \sigma(X_1, \ldots, X_n)$ for a sequence of random variables.
for all $n$. for all $n$.
\end{itemize} \end{itemize}
$(X_n)_n$ is called a \vocab{sub-martingale}, $(X_n)_n$ is called a \vocab{submartingale},
if it is adapted to $\cF_n$ but if it is adapted to $\cF_n$ but
\[ \[
\bE[X_{n+1} | \cF_n] \overset{\text{a.s.}}{\ge} X_n. \bE[X_{n+1} | \cF_n] \overset{\text{a.s.}}{\ge} X_n.
\] \]
It is called a \vocab{super-martingale} It is called a \vocab{supermartingale}
if it is adapted but $\bE[X_{n+1} | \cF_n] \overset{\text{a.s.}}{\le} X_n$. if it is adapted but $\bE[X_{n+1} | \cF_n] \overset{\text{a.s.}}{\le} X_n$.
\end{definition} \end{definition}
\begin{corollary} \begin{corollary}
\label{cor:convexmartingale}
Suppose that $f: \R \to \R$ is a convex function such that $f(X_n) \in L^1(\bP)$. Suppose that $f: \R \to \R$ is a convex function such that $f(X_n) \in L^1(\bP)$.
Suppose that $(X_n)_n$ is a martingale% Suppose that $(X_n)_n$ is a martingale%
\footnote{In this form it means, that there is some filtration, \footnote{In this form it means, that there is some filtration,
that we don't explicitly specify}. that we don't explicitly specify}.
Then $(f(X_n))_n$ is a sub-martingale. Then $(f(X_n))_n$ is a submartingale.
Likewise, if $f$ is concave, then $((f(X_n))_n$ is a super-martingale. Likewise, if $f$ is concave, then $((f(X_n))_n$ is a supermartingale.
\end{corollary} \end{corollary}
\begin{proof} \begin{proof}
Apply \autoref{cjensen}. Apply \autoref{cjensen}.