Let $X$ a positive random variable independent of a Brownian motion $B_t$. Let $M_t=B_{tX}$ and $\mathcal{F_t}=\sigma\{X,B_{s}, s\leq tX\}$. We want to show that M is a $\mathcal{F}_t$-local martingale.
I tried to take the sequence of stopping time $T_n=\inf\{t>0:|B_{t(X+1)}|>n\}$ but I have the problem that X is not independent of $\mathcal{F}_t$.
Do you have any idea?