Is it possible to construct a stochastic process $X\_t$ where the limit


$\lim\_{\Delta \rightarrow 0} \rm{Var}\left(\frac{X\_{t\_0+\Delta}-X\_{t\_0}}{\Delta}\right)$


does not exist but the sample paths are still differentiable?



Differentiability of stochastic process
consider $\sum X\_n e^{int}$ where the $X\_n$ have the property that $X\_n$ are independent and eventually 0. Then every sample path is a trig polynomial and infinitely differentiable, however by arranging that $\sum var(X\_n) < \infty, \sum var(nX\_n) = \infty $ you'll probably get what you want.


