# Right continuous filtration and continuous process

We say that a filtration $$\{\mathcal F_t\}$$ is right-continuous if $$\mathcal F_{t^+}=\mathcal F_t$$ where $$\mathcal F_{t^+}=\bigcap_{s>t}\mathcal F_s.$$

Let $$(X_t)$$ a stochastic process and suppose that $$\mathcal F_t=\sigma \{X_s\mid s\leq t\}$$. Do we have that $$(X_t)$$ is continuous $$\iff$$ $$\{\mathcal F_t\}$$ is continuous ?

If not, could someone explain me in what right-continuous filtration are important ? I'm not sure to really understand why we consider them.

The answer, sadly, is no. There is a marvelous answer to that part here: $\mathcal F_t=\mathcal F_{t^+}$ $\iff$ $(X_t)$ is right continuous
Now, as for why it is interesting to consider the right-continuous augmentation of a filtration: Consider a continuous process $$Y_t$$ and the event that "$$Y$$ closes a loop for the first time". If $$Y$$ is one-dimensional this is really the event that "$$Y$$ turns around". But at time exactly $$t$$, I don't actually know whether I'm turning around (say $$Y$$ is differentiable, we can just see that $$Y'=0$$), but at time $$t+\varepsilon$$, I do know, no matter what $$\varepsilon$$ is (because I can now check whether $$Y'$$has flipped sign in the differentiable case).