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

Fix $0\le r\le t$. Since $M$ is a martingale,Every finite vector consisting of $M_{t+s}-M_t$ and past values $M_{r_1},\dots ,M_{r_n}$ has a multivariate normal distribution. Therefore uncorrelated jointly normal variables are independent, so the increment is independent of every finite vector of past values. A Monotone class theorem then extends this to independence from ${\mathcal{F}}_t=\sigma (M_r:0\le r\le t)$. This is the independent increments of a Gaussian martingale.

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,for all $t\ge 0$ almost surely.