fixed bugs

inputs/lecture_20.tex

@@ -222,7 +222,7 @@ we need the following theorem, which we won't prove here:
     It is also clear that $X^T_n$ is integrable since
     \[
     \bE[|X^T_n|] \le  \sum_{k=1}^{n} \bE[|X_k|] < \infty.
-    \] 
+    \]
 
     We have
     \begin{IEEEeqnarray*}{rCl}
@@ -246,7 +246,7 @@ we need the following theorem, which we won't prove here:
     we get from the above that
     \[
     \bE[X_T] \overset{n \ge M}{=} \bE[X^T_n]  \begin{cases}
-        \le \bE[X_0] & \text{ supermartingale},
+        \le \bE[X_0] & \text{ supermartingale},\\
         = \bE[X_0] & \text{ martingale}.
     \end{cases}
     \]
@@ -259,7 +259,7 @@ we need the following theorem, which we won't prove here:
     Then $\bP[T < \infty] = 1$, but
     \[
     1 = \bE[S_T] \neq  \bE[S_0] = 0.
-    \] 
+    \]
 \end{example}
 
 \begin{theorem}[Optional Stopping]
@@ -269,7 +269,7 @@ we need the following theorem, which we won't prove here:
     taking values in $\N$.
 
     If one of the following holds
-    \begin{itemize}[(i)]
+    \begin{enumerate}[(i)]
         \item $T \le M$ is bounded,
         \item $(X_n)_n$ is uniformly bounded
             and $T < \infty$  a.s.,
@@ -277,7 +277,7 @@ we need the following theorem, which we won't prove here:
             and $|X_n(\omega) - X_{n-1}(\omega)| \le K$
             for all $n \in \N, \omega \in \Omega$ and
             some $K > 0$,
-    \end{itemize}
+    \end{enumerate}
     then $\bE[X_T] \le  \bE[X_0]$.
 
     If $(X_n)_n$ even is a martingale, then