My question: Is the natural filtration of a right-continuous process also right-continuous?


I would say yes, but don't know where to start proving it.


Thanks for your help/ideas!



Right-continuity of natural filtrations
Right continuity fails even for canonical continuous processes. 


The natural filtration on $C([0,\infty))$ is not right continuous. 
For example, the event $\{\omega: {d^+\over dt}\ \omega\_t\mbox{ exists at }t=0\}$ belongs 
to ${\cal F}\_{0+}$ but not ${\cal F}\_0$. In words, you can tell whether
 the function $\omega\_t$ has a right derivative at $t=0$ with an infinitesimal peek *beyond* time 0, 
but you cannot tell just from the value of the function $\omega\_t$ *at* time 0. 


Right continuous filtrations are nicer to work with, and since it fails for the natural filtration,  

we often use the right continuous version instead. Fortunately, many of the nice 
properties of right continuous processes carry over even with this enlarged filtration. For example, 
Brownian motion is still Markov with respect to ${\cal F}\_{t+}$ which leads to interesting results like Blumenthal's zero-one law. 


