# Integrated colored Gaussian noise

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 $Z_t$?
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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 $\int_a^b f(t) dZ_t = \int_a^b f(t) z_t dt.$