Novikov's condition

If a continuous local martingale $M$ satisfiesthen its stochastic exponential is a true martingale through time $T$.

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

Write $M_t=B_{[M{]}_t}$ by the Dambis-Dubins-Schwarz theorem and stop the exponential Brownian martingale when $B_s-s$ first reaches $b<0$. The stopped exponential has mean one. Its boundary contribution before $[M{]}_T$ is at mostwhich vanishes as $b\to -\infty$. The remaining term converges monotonically to $\mathcal{E}(M{)}_t$, proving that its expectation is one.