Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 34 / 4 / a / Solution

The stochastic exponential of $M$ isItô's formula gives $d\mathcal{E}(M{)}_t=\mathcal{E}(M{)}_{t}d{M}_t$, so it is a positive continuous local martingale starting from one. For $p>1$,Every nonnegative local martingale is a supermartingale. Hence, under $[M{]}_{\infty }\le C$,The bound applies also to stopped versions. Localize the exponential to true martingales. Their uniform $L^p$ bound permits $L^1$ passage to the limit, proving that the unstopped exponential is a true martingale of mean one. The same bound makes its entire time-indexed family uniformly integrable. This proves the bounded-bracket criterion for a stochastic exponential, without needing an additional exponential-moment assumption.