  For the class of Gauss-Markov processes we study the problem of asymptotic
equivalence of the nonparametric regression model with errors given by the
increments of the process and the continuous time model, where a whole path of
a sum of a deterministic signal and the Gauss-Markov process can be observed.
In particular we provide sufficient conditions such that asymptotic equivalence
of the two models holds for functions from a given class, and we verify these
for the special cases of Sobolev ellipsoids and H\"older classes with
smoothness index $> 1/2$ under mild assumptions on the Gauss-Markov process at
hand. To derive these results, we develop an explicit characterization of the
reproducing kernel Hilbert space associated with the Gauss-Markov process, that
hinges on a characterization of such processes by a property of the
corresponding covariance kernel introduced by Doob. In order to demonstrate
that the given assumptions on the Gauss-Markov process are in some sense sharp
we also show that asymptotic equivalence fails to hold for the special case of
Brownian bridge. Our results demonstrate that the well-known asymptotic
equivalence of the Gaussian white noise model and the nonparametric regression
model with independent standard normal distributed errors can be extended to a
broad class of models with dependent data.
