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 $Z\_t$?



Integrated colored Gaussian noise
Q1: Rather yes. If we assume that function $w$ is not crazy, e.g. $C^1$, then there exists a continuous version of $z$ and the integral can be computed path-by-path using the Lebesgue (or even Riemann) integral theory.


Q2:Under assumption above $Z$ has paths of the finite variation. Hence there exists the Stieltjes integral (for almost any path) and for any measurable function $f$ one have $\int\_a^b f(t) dZ\_t = \int\_a^b f(t) z\_t dt.$


Q3:I assume that you meant quadratic variation of Z, which is 0.


