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 ?