Let's consider this problem in the book by Karatzas and Shreve.
Problem. Let $\{X_t\}$ be a stochastic process and $\{\mathcal{F}^X_t\}$ its natural filtration.
- Show that the filtration $\mathcal{F}^X_{t+}$ is always right-continuous.
- Show that if $X$ is left-continuous then $\{\mathcal{F}^X_t\}$ is left-continuous too.
- Show by an example that $\{\mathcal{F}^X_t\}$ may fail to be right-continuous even if $X$ is right-continuous.
My problem is with point 1) (the other two are discussed at the end of the post)
Is it true that $$\mathcal F_{t++}=\bigcap_{\varepsilon^{\prime}>0} \mathcal F_{(t+) +\varepsilon^{\prime}}=\bigcap_{\varepsilon^{\prime}>0}\bigcap_{\varepsilon>0}\mathcal F_{t+\varepsilon+\varepsilon^{\prime}}=\bigcap_{\varepsilon>0}\mathcal F_{t+\varepsilon}=\mathcal F_{t+}\quad ?$$
Concerning the point 2) the answer has been given, for example, here: should $X$ be left continuous we would have $X_t=\lim_{n\rightarrow+\infty}X_{t-1/n}$ and since $X_{t-1/n}$ is $\mathcal F_{t-}^X$ measurable we get that $X_t$ is $\mathcal F_{t-}$ measurable too and so $\mathcal F_t^X\subseteq \mathcal F_{t-}^X$ and since the other inclusion $\mathcal F_{t-}^X\subseteq \mathcal F_{t}^X$ is always true we get the $\mathcal F_t^X=\mathcal F_{t-}^X$.
Concerning the last point it is enough to consider, for any continuous process $X$, the event $E_t=\{\omega\in\Omega|X_t(\omega)\textrm{ has a local max/min in }t\}$, so that $E_t\in\mathcal{F}^X_{t+}$ but $E_t\notin\mathcal F^X_{t}$.