Define the deterministic function $f(t)=\mathbb E[M_t^2]$. The independent increments from part (i) show that $f$ is increasing and that $M_t^2-f(t)$ is a <martingale>. Mean-square continuity follows from path continuity and the Gaussian laws, so $f$ is continuous.

The <Itô formula> also says that $M_t^2-[M]_t$ is a <local martingale>. Their difference $[M]_t-f(t)$ is therefore a continuous <finite-variation process> that is also a local martingale. By the theorem that a <continuous finite-variation local martingale is constant>, and because the difference starts at zero,
$$
[M]_t=f(t)
$$
for all $t\geq0$ almost surely.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 202 / 3 / a / ii / Solution

A continuous <local martingale> with <finite variation> is almost surely constant. After localization, its <quadratic variation> is both zero, because it has finite variation, and the quantity controlling its $L^2$ increments, because it is a martingale.

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