Solution
= Solution
For a continuous local martingale $M$ with $M_0=0$, its <stochastic exponential> is
$$
\mathcal E(M)_t=\exp\!\left(M_t-\frac12[M]_t\right).
$$
The <Itô formula> gives $d\mathcal E(M)_t=\mathcal E(M)_t\,dM_t$, so $\mathcal E(M)$ is a positive <continuous local martingale>. Every <nonnegative local martingale> is a <supermartingale>, because localization and the <Conditional Fatou lemma> turn the localized martingale equality into the supermartingale inequality.