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We have given number $1<p< \infty$, $K>0$ and continuous local martingale $M$ such that $\mathbb{E}|M_{\tau}|^p \le K$ for every limited stopping time $\tau$. Show that $M$ is martingale.

Please help me, I have a big problem with solving this task. I don't know how to even start it.

wiwnes691
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    where have you used $p$ (I guess it comes from the $p$-norm though) – user190080 Jun 07 '16 at 12:26
  • Thanks, I ommited it – wiwnes691 Jun 07 '16 at 12:34
  • Do You have any idea how to help me? – wiwnes691 Jun 07 '16 at 17:20
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    it is a rather standard result, just check for example every bounded local martingale is a martingale - you should be able to find a lot of material. – user190080 Jun 07 '16 at 17:33
  • yes you're right: http://www.chiark.greenend.org.uk/~alanb/stoc-calc.pdf - theorem 5.2. But I find it hard to use it in this task and WRITE it down to be properly done – wiwnes691 Jun 07 '16 at 17:55
  • is it just: there exists $\tau_n$ localization sequence such that: $\mathbb{E}(M_t^{\tau_n}|F_s)=M^{\tau_n}s$ so we have $\mathbb{E}(M{t \wedge {\tau_n}}|F_s)=M_{{\tau_n}\wedge s}$ thanks to $\mathbb{E}|M_{\tau}|^p \le K$ it convergences to $\mathbb{E}(M_t|F_s)=M_s$ – wiwnes691 Jun 07 '16 at 18:18

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