Let $a >0$, $(B_t)_{t\geq0}$ be a standard Brownian motion. Define the stopping time $$H_a := \inf\{t \geq 0 : B_t \geq a\}.$$
Then is the martingale $M_t$ where $M_t: = B_{t\wedge H_a}$ bounded in $L^2$?
I'm trying to show that $\mathbb E [ H_a] = +\infty$ and the question says I should use an appropriate characterisation of continuous martingales bounded in $L^2$.