Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 202 / 2 / c

Letting $t\to \infty$ in the Burkholder-Davis-Gundy inequalities with exponent two gives absolute constants $c,C>0$ such thatSince $[X]$ is nondecreasing, $\mathbb{E}[X{]}_{\infty }={\sup }_t\mathbb{E}[X{]}_t$. Thus one of the two quantities in the question is finite exactly when the other is.