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Let $\tau$ be a stopping time for the filtration $\{\mathcal{F}_n\}$ and suppose there is a constant N s.t. for every $n\ge 0$, $\mathcal{P}(\tau\le n+N|\mathcal{F}_n)\ge \epsilon \gt 0$ for some $\epsilon$.

Show that $\mathcal{P}(\tau< \infty)=1$ and $\mathbb{E}[\tau^p]< \infty$ for every $p\ge 1$

  • 0
    I changed your $\le\infty$ to $<\infty$, which I presume is what you intended.2011-11-14
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    Are you sure that you copied the problem correctly? As written, I don't think it is true.2011-11-14
  • 0
    pretty sure its $<\infty$. When I did this question I used the tower property on indicator functions.2012-10-15

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