Infinite quadratic variation forces unbounded oscillations ($[M{]}_{\infty }=\infty ⟹\mathrm{limsup}{M}_t=\infty ,\mathrm{liminf}{M}_t=-\infty$)

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 ${\mathrm{limsup}}_{t\to \infty }|M_t|=\infty$.