I consider two stochastic processes $X_t = B_t$ and $Y_t = B_{t-1}$ with their natural filtrations $\mathcal{F}^X$ and $\mathcal{F}^Y$, where $B_t$ is a Brownian motion.
It is well known that $X_t$ is a martingale with respect to $\mathcal{F}^X$ and $Y_t$ is a martingale with respect to $\mathcal{F}^Y$.
Moreover, It is not difficult to show that $X_t$ is a martingale with respect to $\mathcal{F}^Y$.
My question is: Is $Y_t$ a martingale with respect to $\mathcal{F}^X$? I think that no, but I do not know how to prove it.