Letting $t\to\infty$ in the <Burkholder-Davis-Gundy inequalities> with exponent two gives absolute constants $c,C>0$ such that
$$
c\,\mathbb E[X]_\infty
\leq\mathbb E\sup_{t\geq0}|X_t|^2
\leq C\,\mathbb E[X]_\infty.
$$
Since $[X]$ 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.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-202/2/c/solution