lecture 20
This commit is contained in:
parent
4414f032a8
commit
a9bb7ae3c7
3 changed files with 201 additions and 4 deletions
|
@ -219,7 +219,8 @@ Let $(\Omega, \cF, \bP)$ as always and let $(\cF_n)_n$ always be a filtration.
|
||||||
|
|
||||||
\begin{theorem}
|
\begin{theorem}
|
||||||
\label{ceismartingale}
|
\label{ceismartingale}
|
||||||
Let $X \in L^p$ for some $p \ge 1$.
|
Let $X \in L^p$ for some $p \ge 1$
|
||||||
|
and $\bigcup_n \cF_n \to \cF$.
|
||||||
Then $X_n \coloneqq \bE[X | \cF_n]$ defines a martingale which converges
|
Then $X_n \coloneqq \bE[X | \cF_n]$ defines a martingale which converges
|
||||||
to $X$ in $L^p$.
|
to $X$ in $L^p$.
|
||||||
\end{theorem}
|
\end{theorem}
|
||||||
|
|
|
@ -22,8 +22,8 @@
|
||||||
&\overset{A \in \cF_n}{=}& \lim_{\substack{n \to \infty\\n \ge m}} \bE[X \One_A]\\
|
&\overset{A \in \cF_n}{=}& \lim_{\substack{n \to \infty\\n \ge m}} \bE[X \One_A]\\
|
||||||
\end{IEEEeqnarray*}
|
\end{IEEEeqnarray*}
|
||||||
Hence $\int_A Y \dif \bP = \int_A X \dif \bP$ for all $m \in \N, A \in \cF_m$.
|
Hence $\int_A Y \dif \bP = \int_A X \dif \bP$ for all $m \in \N, A \in \cF_m$.
|
||||||
Since $\cF = \sigma\left( \bigcup \cF_n \right)$
|
Since $\sigma(X) = \bigcup \cF_n$
|
||||||
this holds for all $A \in \cF$.
|
this holds for all $A \in \sigma(X)$.
|
||||||
Hence $X = Y$ a.s., so $X_n \xrightarrow{L^2} X$.
|
Hence $X = Y$ a.s., so $X_n \xrightarrow{L^2} X$.
|
||||||
Since $(X_n)_n$ is uniformly bounded, this also means
|
Since $(X_n)_n$ is uniformly bounded, this also means
|
||||||
$X_n \xrightarrow{L^p} X$.
|
$X_n \xrightarrow{L^p} X$.
|
||||||
|
@ -111,8 +111,203 @@ we need the following theorem, which we won't prove here:
|
||||||
Hence $X_n = \bE[X | \cF_m]$ by the uniqueness of conditional expectation
|
Hence $X_n = \bE[X | \cF_m]$ by the uniqueness of conditional expectation
|
||||||
and by \autoref{ceismartingale},
|
and by \autoref{ceismartingale},
|
||||||
we get the convergence.
|
we get the convergence.
|
||||||
|
|
||||||
\end{refproof}
|
\end{refproof}
|
||||||
|
|
||||||
|
\subsection{Stopping times}
|
||||||
|
|
||||||
|
\begin{definition}[Stopping time]
|
||||||
|
A random variable $T: \Omega \to \N \cup \{\infty\}$ on a filtered probability space $(\Omega, \cF, \{\cF_n\}_n, \bP)$ is called a \vocab{stopping time},
|
||||||
|
if
|
||||||
|
\[
|
||||||
|
\{T \le n\} \in \cF_n
|
||||||
|
\]
|
||||||
|
for all $n \in \N$.
|
||||||
|
Equivalently, $\{T = n\} \in \cF_n$ for all $n \in \N$.
|
||||||
|
|
||||||
|
\end{definition}
|
||||||
|
|
||||||
|
\begin{example}
|
||||||
|
A constant random variable $T = c$ is a stopping time.
|
||||||
|
\end{example}
|
||||||
|
|
||||||
|
\begin{example}[Hitting times]
|
||||||
|
For an adapted process $(X_n)_n$
|
||||||
|
with values in $\R$ and $A \in \cB(\R)$, the \vocab{hitting time}
|
||||||
|
\[
|
||||||
|
T \coloneqq \inf \{n \in \N : X_n \in A\}
|
||||||
|
\]
|
||||||
|
is a stopping time,
|
||||||
|
as
|
||||||
|
\[
|
||||||
|
\{T \le n \} = \bigcup_{k=1}^n \{X_k \in A\} \in \cF_n.
|
||||||
|
\]
|
||||||
|
|
||||||
|
However, the last exit time
|
||||||
|
\[
|
||||||
|
T \coloneqq \sup \{n \in \N : X_n \in A\}
|
||||||
|
\]
|
||||||
|
is not a stopping time.
|
||||||
|
|
||||||
|
\end{example}
|
||||||
|
|
||||||
|
|
||||||
|
\begin{example}
|
||||||
|
Consider the simple random walk, i.e.
|
||||||
|
$X_n$ i.i.d.~with $\bP[X_n = 1] = \bP[X_n = -1] = \frac{1}{2}$.
|
||||||
|
Set $S_n \coloneqq \sum_{i=1}^{n} X_n$.
|
||||||
|
Then
|
||||||
|
\[
|
||||||
|
T \coloneqq \inf \{n \in \N : S_n \ge A \lor S_n \le B\}
|
||||||
|
\]
|
||||||
|
is a stopping time.
|
||||||
|
\end{example}
|
||||||
|
|
||||||
|
\begin{example}
|
||||||
|
If $T_1, T_2$ are stopping times with respect to the same filtration,
|
||||||
|
then
|
||||||
|
\begin{itemize}
|
||||||
|
\item $T_1 + T_2$,
|
||||||
|
\item $\min \{T_1, T_2\}$ and
|
||||||
|
\item $\max \{T_1, T_2\}$
|
||||||
|
\end{itemize}
|
||||||
|
are stopping times.
|
||||||
|
|
||||||
|
Note that $T_1 - T_2$ is not a stopping time.
|
||||||
|
|
||||||
|
\end{example}
|
||||||
|
|
||||||
|
\begin{remark}
|
||||||
|
There are two ways to interpret the interaction between a stopping time $T$
|
||||||
|
and a stochastic process $(X_n)_n$.
|
||||||
|
\begin{itemize}
|
||||||
|
\item The behaviour of $ X_n$ until $T$,
|
||||||
|
i.e.~looking at the \vocab{stopped process}
|
||||||
|
\[
|
||||||
|
X^T \coloneqq \left(X_{T \wedge n}\right)_{n \in \N}
|
||||||
|
\].
|
||||||
|
\item The value of $(X_n)_n)$ at time $T$,
|
||||||
|
i.e.~looking at $X_T$.
|
||||||
|
\end{itemize}
|
||||||
|
\end{remark}
|
||||||
|
\begin{example}
|
||||||
|
If we look at a process
|
||||||
|
\[
|
||||||
|
S_n = \sum_{i=1}^{n} X_i
|
||||||
|
\]
|
||||||
|
for some $(X_n)_n$, then
|
||||||
|
\[
|
||||||
|
S^T = (\sum_{i=1}^{T \wedge n} X_i)_n
|
||||||
|
\]
|
||||||
|
and
|
||||||
|
\[
|
||||||
|
S_T = \sum_{i=1}^{T} X_i.
|
||||||
|
\]
|
||||||
|
\end{example}
|
||||||
|
|
||||||
|
\begin{theorem}
|
||||||
|
If $(X_n)_n$ is a supermartingale and $T$ is a stopping time,
|
||||||
|
then $X^T$ is also a supermartingale,
|
||||||
|
and we have $\bE[X_{T \wedge n}] \le \bE[X_0]$ for all $n$.
|
||||||
|
If $(X_n)_n$ is a martingale, then so is $X^T$
|
||||||
|
and $\bE[X_{T \wedge n}] \le \bE[X_0]$.
|
||||||
|
\end{theorem}
|
||||||
|
\begin{proof}
|
||||||
|
First, we need to show that $X^T$ is adapted.
|
||||||
|
This is clear since
|
||||||
|
\begin{IEEEeqnarray*}{rCl}
|
||||||
|
X^T_n &=& X_T \One_{T < n} + X_n \One_{T \ge n}\\
|
||||||
|
&=& \sum_{k=1}^{n-1} X_k \One_{T = k} + X_n \One_{T \ge n}.
|
||||||
|
\end{IEEEeqnarray*}
|
||||||
|
|
||||||
|
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}
|
||||||
|
\bE[X^T_n - X^T_{n-1} | \cF_{n-1}]
|
||||||
|
&=& \bE[X_n \One_{\{T \ge n\}} + \sum_{k=1}^{n-1} X_k \One_{\{ T = k\} } - X_{n-1}(\One_{T \ge n} + \One_{\{T = n-1\}})
|
||||||
|
+ \sum_{k=1}^{n-2} X_k \One_{\{T = k\} } | \cF_{n-1}]\\
|
||||||
|
&=& \bE[(X_n - X_{n-1}) \One_{\{ T \ge n\} } | \cF_{n-1}]\\
|
||||||
|
&=& \One_{\{ T \ge n\}} (\bE[X_n | \cF_{n-1}] - X_{n-1})\\
|
||||||
|
&& \begin{cases}
|
||||||
|
\le 0\\
|
||||||
|
= 0 \text{ if $(X_n)_n$ is a martingale}.
|
||||||
|
\end{cases}.
|
||||||
|
\end{IEEEeqnarray*}
|
||||||
|
|
||||||
|
\end{proof}
|
||||||
|
|
||||||
|
\begin{remark}
|
||||||
|
\label{roptionalstoppingi}
|
||||||
|
We now want a similar statement for $X_T$.
|
||||||
|
In the case that $T \le M$ is bounded,
|
||||||
|
we get from the above that
|
||||||
|
\[
|
||||||
|
\bE[X_T] \overset{n \ge M}{=} \bE[X^T_n] \begin{cases}
|
||||||
|
\le \bE[X_0] & \text{ supermartingale},
|
||||||
|
= \bE[X_0] & \text{ martingale}.
|
||||||
|
\end{cases}
|
||||||
|
\]
|
||||||
|
|
||||||
|
However if $T$ is not bounded, this does not hold in general.
|
||||||
|
\end{remark}
|
||||||
|
\begin{example}
|
||||||
|
Let $(S_n)_n$ be the simple random walk
|
||||||
|
and take $T \coloneqq \inf \{n : S_n = 1\}$.
|
||||||
|
Then $\bP[T < \infty] = 1$, but
|
||||||
|
\[
|
||||||
|
1 = \bE[S_T] \neq \bE[S_0] = 0.
|
||||||
|
\]
|
||||||
|
\end{example}
|
||||||
|
|
||||||
|
\begin{theorem}[Optional Stopping]
|
||||||
|
\label{optionalstopping}
|
||||||
|
Let $(X_n)_n$ be a supermartingale
|
||||||
|
and let $T$ be a stopping time
|
||||||
|
taking values in $\N$.
|
||||||
|
|
||||||
|
If one of the following holds
|
||||||
|
\begin{itemize}[(i)]
|
||||||
|
\item $T \le M$ is bounded,
|
||||||
|
\item $(X_n)_n$ is uniformly bounded
|
||||||
|
and $T < \infty$ a.s.,
|
||||||
|
\item $\bE[T] < \infty$
|
||||||
|
and $|X_n(\omega) - X_{n-1}(\omega)| \le K$
|
||||||
|
for all $n \in \N, \omega \in \Omega$ and
|
||||||
|
some $K > 0$,
|
||||||
|
\end{itemize}
|
||||||
|
then $\bE[X_T] \le \bE[X_0]$.
|
||||||
|
|
||||||
|
If $(X_n)_n$ even is a martingale, then
|
||||||
|
under the same conditions
|
||||||
|
$\bE[X_T] = \bE[X_0]$.
|
||||||
|
\end{theorem}
|
||||||
|
\begin{proof}
|
||||||
|
(i) was dealt with in \autoref{roptionalstoppingi}.
|
||||||
|
|
||||||
|
(ii): Since $(X_n)_n$ is bounded, we get that
|
||||||
|
\begin{IEEEeqnarray*}{rCl}
|
||||||
|
\bE[|X_T - X_0|] &\overset{\text{dominated convergence}}{=}& \lim_{n \to \infty} \bE[|X_{T \wedge n} - X_0|]\\
|
||||||
|
&\overset{\text{part (i)}}{\le}& 0.
|
||||||
|
\end{IEEEeqnarray*}
|
||||||
|
|
||||||
|
(iii): It is
|
||||||
|
\begin{IEEEeqnarray*}{rCl}
|
||||||
|
|X_{T \wedge n}- X_0| &\le& | \sum_{k=1}^{T \wedge n} X_k - X_{k-1}|\\
|
||||||
|
&\le & (T \wedge n) \cdot K\\
|
||||||
|
&\le & T \cdot K < \infty.
|
||||||
|
\end{IEEEeqnarray*}
|
||||||
|
|
||||||
|
Hence, we can apply dominated convergence and obtain
|
||||||
|
\begin{IEEEeqnarray*}{rCl}
|
||||||
|
\bE[X_T - X_0] &=& \lim_{n \to \infty} \bE[X_{T \wedge n} - X_0].
|
||||||
|
\end{IEEEeqnarray*}
|
||||||
|
Thus, we can apply (ii).
|
||||||
|
|
||||||
|
|
||||||
|
The statement about martingales follows from
|
||||||
|
applying this to $(X_n)_n$ and $(-X_n)_n$,
|
||||||
|
which are both supermartingales.
|
||||||
|
\end{proof}
|
||||||
|
|
|
@ -43,6 +43,7 @@
|
||||||
\input{inputs/lecture_17.tex}
|
\input{inputs/lecture_17.tex}
|
||||||
\input{inputs/lecture_18.tex}
|
\input{inputs/lecture_18.tex}
|
||||||
\input{inputs/lecture_19.tex}
|
\input{inputs/lecture_19.tex}
|
||||||
|
\input{inputs/lecture_20.tex}
|
||||||
|
|
||||||
\cleardoublepage
|
\cleardoublepage
|
||||||
|
|
||||||
|
|
Loading…
Reference in a new issue