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!
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 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.