# Differentiability of stochastic process

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?

• You want an example where the variance always exists, right? Otherwise, you could take a r.v. $\xi$ with nonexistent 2nd moment, and let $X_t=\xi t$. – Serguei Popov Nov 20 '15 at 10:23
• Thanks @Serguei. Yes, for every $t$, the variance of the random variable $X_t$ should be defined. – mcsanchez Nov 21 '15 at 4:49

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.
• Thanks @Michael. It seems like Gaussians with mean $1/n$ and variance $1/n^2$ would work. – mcsanchez Nov 21 '15 at 5:09