TITLE: Does there exist an almost surely differentiable martingale?
QUESTION [5 upvotes]: Does there exist a continuous time martingale $X_t$ not a.s. constant in $t$ that is almost surely everywhere differentiable?

REPLY [2 votes]: Others have answered that there does not exist an a.s. differentiable martingale process that's not constant. One related fact (not asked for but interesting) is that an a.s. differentiable Markov process satisfies a deterministic ODE with random initial conditions. See https://www.ams.org/journals/notices/196808/196808FullIssue.pdf page 748
Somehow martingale/Markov don't work well with differentiability.