Suppose $Y$ is a local martingale started from $0$ and define for each $k \gt 0$ the stopping time $T_k=\inf\{t \gt 0 : |Y_t|=k\}$. Then, for a given $k$, $(Y^{T_k}_t)_{t\geq 0}$ is a bounded continuous martingale. Is it true that it is also an $L^2$ martingale? I think it is, and in fact that it is in any $L^p$ but I would like to have confirmation. I mean, why can't I just write
$$ E[(Y^{T_k}_t)^2] \leq E[k^2]=k^2$$