Theorem
For any two increasing functions $f,g:\mathbb{R}\to\mathbb{R}$ with finite $\mathbb{E}[f(X)]<\infty$ and $\mathbb{E}[g(X)]<\infty$,
$$ \mathrm{cov}[f(X), g(X)]\geq 0 $$
The above basic theorem can be found in e.g., 1. But it is used when $X$ is a random variable.
My question is what about $X(t)$ is a process? That is, if $f$ and $g$ are chosen such that $f(X(t))$ and $g(X(t))$ are almost surely increasing processes, will $ \mathrm{cov}[f(X(t)), g(X(t))]\geq 0 $ hold for any $t>0$?