 Solution

The assertion is false. Let $Z\sim N(0,1)$ and define $X_t=tZ$. This is a centered continuous Gaussian process. Its natural filtration satisfies $Z=X_s/s\in {\mathcal{F}}_s^X$ for every $s>0$, and hence, for $0<s<t$,with positive probability. Thus $X$ is not a martingale and does not belong to the stated martingale class.