My question: Is the natural filtration of a rightcontinuous process also rightcontinuous?
I would say yes, but don't know where to start proving it.
Thanks for your help/ideas!
My question: Is the natural filtration of a rightcontinuous process also rightcontinuous? I would say yes, but don't know where to start proving it. Thanks for your help/ideas! 


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, 

