Dambis-Dubins-Schwarz proof of the Novikov condition
= Dambis-Dubins-Schwarz proof of the Novikov condition
{c}
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 most
$$
e^b\mathbb E e^{[M]_T/2},
$$
which vanishes as $b\to-\infty$. The remaining term converges monotonically to $\mathcal E(M)_t$, proving that its expectation is one.