= Infinite quadratic variation forces unbounded oscillations
{title2=$[M]_\infty=\infty\Longrightarrow\limsup M_t=\infty,\ \liminf M_t=-\infty$}
For a <continuous local martingale> with infinite terminal <quadratic variation>, the <Dambis-Dubins-Schwarz theorem> represents $M_t-M_0=W_{[M]_t}$. Its continuous clock traverses all nonnegative times. Standard one-dimensional <Brownian motion> has unbounded positive and negative oscillations, by the <Brownian reflection principle> or <recurrence of one-dimensional Brownian motion>. Continuity bounds every path on compact time intervals, so these large values occur at arbitrarily late times. Pulling them back through the clock proves the displayed conclusion <almost surely>, and in particular $\limsup_{t\to\infty}|M_t|=\infty$.
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