Divergent logarithmic clock for a positive local martingale tending to zero
= Divergent logarithmic clock for a positive local martingale tending to zero
{title2=$M_t\to0\ \Longrightarrow\ \langle X\rangle_\infty=\infty$}
Write a positive continuous local martingale as $M=\exp(X-\langle X\rangle/2)$ using its <stochastic logarithm>. If $M_t\to0$, its logarithmic bracket must diverge. Otherwise the <finite-bracket convergence lemma> makes $X$ converge finitely and the exponential has a positive limit.