Right-continuity of natural filtrations - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T17:53:09Z http://mathoverflow.net/feeds/question/46957 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/46957/right-continuity-of-natural-filtrations Right-continuity of natural filtrations Johannes 2010-11-22T14:55:14Z 2010-11-22T15:46:34Z <p>My question: Is the natural filtration of a right-continuous process also right-continuous?</p> <p>I would say yes, but don't know where to start proving it.</p> <p>Thanks for your help/ideas!</p> http://mathoverflow.net/questions/46957/right-continuity-of-natural-filtrations/46962#46962 Answer by Byron Schmuland for Right-continuity of natural filtrations Byron Schmuland 2010-11-22T15:46:34Z 2010-11-22T15:46:34Z <p>Right continuity fails even for canonical continuous processes. </p> <p>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 <em>beyond</em> time 0, but you cannot tell just from the value of the function $\omega_t$ <em>at</em> time 0. </p> <p>Right continuous filtrations are nicer to work with, and since it fails for the natural filtration,<br> 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. </p>