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@ -74,6 +74,16 @@ The Radon Nikodym theorem is the converse of that:
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w.r.t.~$\mu$.
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This is written as $\nu \ll \mu$.
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\end{definition}
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\begin{definition}+
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Two measures $\mu$ and $\nu$ on a measure space $(\Omega, \cF)$
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are called \vocab{singular},
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denoted $\mu \bot \nu$,
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iff there exists $A \in \cF$ such that
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\[
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\mu(A) = \nu(A^c) = 0.
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\]
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\end{definition}
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With \autoref{radonnikodym} we get a very short proof of the existence
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of conditional expectation:
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@ -171,6 +181,7 @@ we have gathered up to time $n$.
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Typically $\cF_n = \sigma(X_1, \ldots, X_n)$ for a sequence of random variables.
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\begin{definition}
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\label{def:martingale}
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Let $(\cF_n)$ be a filtration and
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$X_1,\ldots,X_n$ be random variables such that $X_i \in L^1(\bP)$.
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Then we say that $(X_n)_{n \ge 1}$ is an $(\cF_n)_n$-\vocab{martingale}
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@ -20,6 +20,7 @@ Hence the same holds for submartingales, i.e.
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\end{question}
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\begin{example}[A martingale not converging in $L^1$ ]
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\label{ex:martingale-not-converging-in-l1}
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Fix $u > 1$ and let $p = \frac{1}{1+u}$.
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Let $ (Z_n)_{n \ge 1}$ be i.i.d.~$\pm 1$ with
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$\bP[Z_n = 1] = p$.
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@ -23,6 +23,7 @@
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\end{goal}
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\begin{definition}
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\label{def:ui}
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A sequence of random variables $(X_n)_n$ is called \vocab{uniformly integrable} (UI),
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if
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\[\forall \epsilon > 0 .~\exists K > 0 .~ \forall n.~
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@ -245,4 +246,5 @@ Let $(\Omega, \cF, \bP)$ as always and let $(\cF_n)_n$ always be a filtration.
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Let $(X_n)_n$ be a martingale bounded in $L^p$.
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Then there exists a random variable $X \in L^p$, such that
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$X_n = \bE[X | \cF_n]$ for all $n$.
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In particular, $X_n \xrightarrow{L^p} X$.
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\end{theorem}
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@ -117,7 +117,7 @@ In this lecture we recall the most important point from the lecture.
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\begin{itemize}
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\item iid (\autoref{clt}),
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\item Lindeberg (\autoref{lindebergclt}),
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\item Luyapanov (\autoref{lyapunovclt})
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\item Lyapanov (\autoref{lyapunovclt})
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\end{itemize}
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\item How to apply this? Exercises!
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\end{itemize}
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@ -125,10 +125,12 @@ In this lecture we recall the most important point from the lecture.
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\subsubsection{Conditional expectation}
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\begin{itemize}
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\item Definition and existence of conditional expectation for $X \in L^1(\Omega, \cF, \bP)$
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(\autoref{conditionalexpectation})
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\item If $H = L^2(\Omega, \cF, \bP)$, then $\bE[ \cdot | \cG]$
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is the (unique) projection on the closed subspace $L^2(\Omega, \cG, \bP)$.
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Why is this a closed subspace? Why is the projection orthogonal?
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\item Radon-Nikodym Theorem (Proof not relevant for the exam)
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\item Radon-Nikodym Theorem \ref{radonnikodym}
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(Proof not relevant for the exam)
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\item (Non-)examples of mutually absolutely continuous measures
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Singularity in this context? % TODO
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\end{itemize}
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@ -136,25 +138,31 @@ In this lecture we recall the most important point from the lecture.
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\subsubsection{Martingales}
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\begin{itemize}
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\item Definition of Martingales
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\item Doob's convergence theorem, Upcrossing inequality
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\item Definition of Martingales (\autoref{def:martingale})
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\item Doob's convergence theorem (\autoref{doobmartingaleconvergence}),
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Upcrossing inequality (\autoref{lec17l1}, \autoref{lec17l2}, \autoref{lec17l3})
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(downcrossings for submartingales)
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\item Examples of Martingales converging a.s.~but not in $L^1$
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\item Bounded in $L^2$ $\implies$ convergence in $L^2$.
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(\autoref{ex:martingale-not-converging-in-l1})
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\item Bounded in $L^2$ $\implies$ convergence in $L^2$
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(\autoref{martingaleconvergencel2}).
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\item Martingale increments are orthogonal in $L^2$!
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(\autoref{martingaleincrementsorthogonal})
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\item Doob's (sub-)martingale inequalities
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(\autoref{dooblp}),
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\item $\bP[\sup_{k \le n} M_k \ge x]$ $\leadsto$ Look at martingale inequalities!
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Estimates might come from Doob's inequalities if $(M_k)_k$ is a (sub-)martingale.
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\item Doob's $L^p$ convergence theorem.
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\item Doob's $L^p$ convergence theorem
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(\autoref{ceismartingale}, \autoref{martingaleisce}).
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\begin{itemize}
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\item Why is $p > 1$ important? \textbf{Role of Banach-Alaoglu}
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\item This is an important proof.
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\end{itemize}
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\item Uniform integrability % TODO
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\item What are stopping times? \autoref{def:stopping-time}
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\item Uniform integrability (\autoref{def:ui})
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\item What are stopping times? (\autoref{def:stopping-time})
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\item (Non-)examples of stopping times
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\item \textbf{Optional stopping theorem} - be really comfortable with this.
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\autoref{optionalstopping}
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\item \textbf{Optional stopping theorem} (\autoref{optionalstopping})
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- be really comfortable with this.
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\end{itemize}
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