  In this work, we study the class of stochastic process that generalizes the
Ornstein-Uhlenbeck processes, hereafter called by \emph{Generalized
Ornstein-Uhlenbeck Type Process} and denoted by GOU type process. We consider
them driven by the class of noise processes such as Brownian motion, symmetric
$\alpha$-stable L\'evy process, a L\'evy process, and even a Poisson process.
We give necessary and sufficient conditions under the memory kernel function
for the time-stationary and the Markov properties for these processes. When the
GOU type process is driven by a L\'evy noise we prove that it is infinitely
divisible showing its generating triplet. Several examples derived from the GOU
type process are illustrated showing some of their basic properties as well as
some time series realizations. These examples also present their theoretical
and empirical autocorrelation or normalized codifference functions depending on
whether the process has a finite or infinite second moment. We also present the
maximum likelihood estimation as well as the Bayesian estimation procedures for
the so-called \emph{Cosine process}, a particular process in the class of GOU
type processes. For the Bayesian estimation method, we consider the power
series representation of Fox's H-function to better approximate the density
function of a random variable $\alpha$-stable distributed. We consider four
goodness-of-fit tests for helping to decide which \emph{Cosine process} (driven
by a Gaussian or an $\alpha$-stable noise) best fit real data sets. Two
applications of GOU type model are presented: one based on the Apple company
stock market price data and the other based on the cardiovascular mortality in
Los Angeles County data.
