Proof or disproof that $Y_t = B_{t-1}$, $t\geq1$ is a martingale with respect to $\mathcal{F}^X$, where $\mathcal{F}^X$ is a natural filtration of $X_t = B_t$.
It is obvious that $Y_t$ is adapted to $\mathcal{F}^X$, because $Y_t$ is adapted to $\mathcal{F}^Y$ and $\mathcal{F}^Y\subset \mathcal{F}^X$, so $Y_t$ is adapted to $\mathcal{F}^X$.
I do not know how to show that $\mathbb{E}[Y_t|\mathcal{F}^X_s] = Y_s$ (or show that it does not hold).
I suppose that I have to use some property of conditional expectation, but I do not know how to do it.
Let $t< s <t+1$ Then, $\mathbb E (B_{s-1}|\mathcal F_t)=B_{s-1}$ since $B_{s-1}$ is already $\mathcal F_t -$measurable. So $\mathbb E (B_{s-1}|\mathcal F_t)\neq B_{t-1}$.