29 lines
648 B
TeX
29 lines
648 B
TeX
\section{(Counter)examples}
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Consistent families and inconsistent families
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Notions of convergence
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Exercise 4.3
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10.2
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\begin{example}[Martingale not converging in $L^1$]
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Let $\Omega = [0,1]$, $\bP = \lambda\upharpoonright [0,1]$.
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Define $X_n \coloneqq 2^n \cdot \One_{[0,2^n]}$,
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and let $(\cF_n)_n$ be the canonical filtration.
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Then $(X_n)_{n}$ is a Martingale
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with $\bE[X_0] = 1$,
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but $X_n \xrightarrow{a.s.} 0$.
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\end{example}
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Stopping times
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\begin{example}[{Martingale such that $\bE[X_T] \neq \bE[X_0]$}]
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Consider the simple random walk and $T = \inf \{n : X_n \ge 1\}$.
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Obviously $X_T = 1$.
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\end{example}
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