This is exercise 3.28 of Brownian motion, Martingales, and Stochastic Calculus, Le Gall.
$B_t$ be a Standard BM and a>0.
Define $\sigma_a = \inf \{t \geq 0 : B_t \leq t-a \} $ and let $\mu$ in $\mathbb{R}$ and $M_t = \exp( \mu B_t-\mu^2 t /2 )$.
Then show that the stopped martingale $M_{\sigma_a \wedge t }$ is closed iff $\mu \leq 1 $.
(Hint : This martingale is closed iff $\mathbb{E}[M_{\sigma_a}]=1 $)
I know the fact that $\mathbb{E} [\exp (\lambda\sigma_a )] = \exp(-a (\sqrt {1+2 \lambda }-1 ))$, but I can't do any more....
Appreciate any hints.