Solution
= Solution
The <Doléans-Dade exponential>
$$
X_t=\exp\left(M_t-\frac12[M]_t\right)
$$
is a positive continuous local martingale with $X_0=1$, and part c shows $X_t\to0$. Apply part a with $a=e^y$:
$$
\boxed{
\mathbb P\left(\sup_{t\geq0}\left\{M_t-\frac12[M]_t\right\}>y\right)
=\mathbb P(\sup_tX_t>e^y)=e^{-y}.}
$$