Assume we have a colored Gaussian process $z_t$, with an autocorrelation function $cov(z_t,z_s)$ given by an analytical function $\alpha(t,s)$ (if it helps, one can assume that $\alpha(t,s) = \kappa e^{-w(t-s)}$). Consider now a process defined by

$Z_t := \int_0^t z_s ds$

Now, my 3 questions:

- is $Z_t$ well-defined?
- if so, is $d Z_t = z_t dt$ true in any sense?
- what is the quadratic variation of $Z_t$?