Solution
= 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 $0<s<t$,
$$
\mathbb E[X_t\mid\mathcal F_s^X]=tZ=\frac tsX_s\ne X_s
$$
with positive probability. Thus $X$ is not a <martingale> and does not belong to the stated martingale class.