Bounded-bracket criterion for a stochastic exponential
= Bounded-bracket criterion for a stochastic exponential
{title2=$[M]_\infty\leq C$}
For a zero-start <continuous local martingale> $M$ with $[M]_\infty\leq C$ deterministically, its <stochastic exponential> is a <uniformly integrable> <martingale>. For each $p>1$, $\sup_t\mathbb E\mathcal E(M)_t^p\leq e^{p(p-1)C/2}$. Its terminal value is strictly positive, so it defines an equivalent infinite-horizon <change of measure> by the <Girsanov theorem>.