union of natural filtration vs union of right continuous filtration

Question

Assume for a given rcll $\mathbb{R}$-valued process $X=(X_s)_{s\geq 0}$ we define $\tilde{\mathcal{F}}_t=\sigma(X_s:s\leq t)$ and $\mathcal{F}_t=\bigcap_{s>t}\sigma(X_u:u\leq s)$. (I made the assumptions for $X$, just for the case, that there are some complex examples, where they may not coincide). Can we say, that $\tilde{\mathcal{F}}=\bigcup_{t\geq 0}\tilde{\mathcal{F}}_t$ and $\mathcal{F}=\bigcup_{t\geq 0}\mathcal{F}_t$ coincide? So far i dont see any problems.

Answer 1

$\mathcal{F}_t\subset\tilde{\mathcal{F}}_{t+\delta}$ for any $\delta>0$, so $ \bigcup_{t\geq 0}\mathcal{F}_t\subset \bigcup_{t\geq 0}\tilde{\mathcal{F}}_t$, and conversely the right-continuous filtration always contains the natural filtration. You don't need the rcll assumption.