If a continuous local martingale starts at zero and has deterministic continuous <quadratic variation> $[M]_t=f(t)$, the <Dambis-Dubins-Schwarz theorem> gives $M_t=B_{f(t)}$. It is therefore a centered <Gaussian process>. The converse follows from <independent increments of a Gaussian martingale>.

Deterministic quadratic variation characterizes a Gaussian continuous local martingale

Fix $0\leq r\leq t$. Since $M$ is a <martingale>,
$$
\operatorname{Cov}(M_{t+s}-M_t,M_r)=\mathbb E\!\left[M_r\,\mathbb E[M_{t+s}-M_t\mid\mathcal F_t]\right]=0.
$$
Every finite vector consisting of $M_{t+s}-M_t$ and past values $M_{r_1},\ldots,M_{r_n}$ 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 $\mathcal F_t=\sigma(M_r:0\leq r\leq t)$. This is the <independent increments of a Gaussian martingale>.

Solution

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

A centered continuous <Gaussian process> that is also a <martingale> has independent increments. Every future increment has zero <covariance> with every finite vector of past values by the martingale property; <uncorrelated jointly normal variables are independent>.

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 Incoming links: Independent increments of a Gaussian martingale