The <L2 martingale convergence theorem> gives $M_t\to M_\infty$ in $L^2$, so
$$
\|M_\infty\|_2\leq
\left\|\sup_{t\geq0}|M_t|\right\|_2.
$$
Conversely, apply the <Doob L2 maximal inequality> on $[0,T]$:
$$
\mathbb E\sup_{t\leq T}|M_t|^2
\leq4\mathbb E|M_T|^2
\leq4\mathbb E|M_\infty|^2.
$$
<Monotone convergence theorem>[Monotone convergence] as $T\to\infty$ gives
$$
\left\|\sup_{t\geq0}|M_t|\right\|_2
\leq2\|M_\infty\|_2.
$$
Thus the two norms are equivalent.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 202 / 1 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-202/1/a/solution