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 $a=e^y$:
$$
\boxed{
\mathbb P\left(\sup_{t\geq0}\left\{M_t-\frac12[M]_t\right\}>y\right)
=\mathbb P(\sup_tX_t>e^y)=e^{-y}.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 202 / 4 / d / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-202/4/d/solution