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Let X be a positive random variable independent of a standard Brownian motion B. Let $M_t = B_{tX}$ for t > 0. We assume that the random variable X is $F_t$ measurable for all t $\geq$ 0, require to show: $M_t$ is adapted to the filtration $(F_t)$.

The question doesn't tell me what $(F_t)$. I guess it is the filtration generated by B ?

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When filtration isn't specified, it is generaly assumed to be the one generated by the stochastic process itself –  Jean-Sébastien Oct 16 '12 at 20:47
    
What is your question? –  Did Oct 17 '12 at 6:52
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If $\mathcal F_t=\sigma(B_s;0\leqslant s\leqslant t)$, this is obviously false: take $X=2$ with full probability, then $M_t=B_{2t}$ is not measurable with respect to $\mathcal F_t$ (except when $t=0$).

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I understand. Thank you for your answer! –  XXX11235 Dec 5 '12 at 2:40
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