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?
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Sign up to join this communityIs 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?
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.