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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:
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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 $\int_a^b f(t) dZ_t = \int_a^b f(t) z_t dt.$ Q3:I assume that you meant quadratic variation of Z, which is 0. |
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