= Solution
The <stochastic exponential> of $M$ is
$$
\boxed{\mathcal E(M)_t=\exp(M_t-\tfrac12[M]_t).}
$$
Itô's formula gives $d\mathcal E(M)_t=\mathcal E(M)_t\,dM_t$, so it is a positive continuous local <martingale> starting from one. For $p>1$,
$$
\mathcal E(M)_t^p=\mathcal E(pM)_t\exp\left(\tfrac12p(p-1)[M]_t\right).
$$
Every nonnegative local <martingale> is a supermartingale. Hence, under $[M]_\infty\leq C$,
$$
\boxed{\sup_t\mathbb E\mathcal E(M)_t^p\leq e^{p(p-1)C/2}.}
$$
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.
Back to article page
