What is the intuition behind right-continuous filtration? I cannot understand the concept of it.
So a filtration is right continuous if for every $t$ it holds that:
$\mathcal{F_t}=\bigcap\limits_{\varepsilon>0}\mathcal{F_{t+\varepsilon}}$
But if for every $t$, then it also holds for $t=0$. And if I choose a large $\epsilon$, then it means that at time zero I know every information about the process?
 A: The idea is that you gain no additional information by taking an infinitesimal step forward in time.
Remember that an $\mathit{intersection}$ means that we are taking only the elements contained in EVERY set in the intersection.  So, if we think of each $F_t$ as the information contained in the system up to time $t$, the intersection $\cap_{\epsilon > 0} \mathcal{F}_{t+\epsilon}$ contains only the information in EVERY $\mathcal{F}_{t+\epsilon}$ for every possible value of $\epsilon > 0$.  That is, only the information contained up until $t+\epsilon$ for every $\epsilon > 0$, in particular, any arbitrarily small $\epsilon$.  So, in this intersection, we have added only the information gained by taking an infinitesimally small step forward in time.
Thus, the idea of right continuity, $\mathcal{F}_t=\cap_{\epsilon > 0} \mathcal{F}_{t+\epsilon}$ is that no information is added in this infinitesimal step.  In other words, there are no instantaneous developments of the system, it evolves in a continuous fashion going forward in time.
(Much credit for this answer is due to Huyen Pham, whose book I'm currently using to review some of this material.)
