# 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?

• No, because you're taking the intersection over $\varepsilon>0$. (To avoid the problem of uncountable intersections we can assume WLOG that $\varepsilon$ is rational.) – Math1000 Jul 14 '16 at 14:56
• @Math1000 +1 for the comment, but I don't see any "problem" with uncountable intersections. – user940 Jul 14 '16 at 15:55
• @Math1000 Okay but what is the intuition behind it? why is it useful? – FelB Jul 14 '16 at 17:20
• @ByronSchmuland Good point, as we are taking the intersection of $\sigma$-algebras as opposed to the intersections of elements of $\sigma$-algebras... – Math1000 Jul 14 '16 at 20:45

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.