Assume we work on a filtered probability space $(\Omega, \mathscr{F},(\mathscr{F})_{t\geq0}, \mathbb{P} )$
Let $B_t$ be a standard $\mathscr{F}_t$-Brownian motion and let $X$ be a positive random variable measurable with respect to $\mathscr{F}_0$. How do I show that $M_t = B_{Xt}$ is a local $\mathscr{F}_{Xt}$-Martingale and a $\mathscr{F}_{Xt}$-Martingale iff $\mathbb{E}\sqrt X < \infty$?
Writing $$M_t=\sqrt X \frac{1}{\sqrt X} B_{\frac{t}{(\sqrt X)^{-2}}} = \sqrt X \hat B^x_t$$ seems to be a step in a right direction. But I am not sure how to proceed from there.