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In this work, we study the class of stochastic process that generalizes theOrnstein-Uhlenbeck processes, hereafter called by \emph{GeneralizedOrnstein-Uhlenbeck Type Process} and denoted by GOU type process. We considerthem 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 functionfor the time-stationary and the Markov properties for these processes. When theGOU type process is driven by a L\'evy noise we prove that it is infinitelydivisible showing its generating triplet. Several examples derived from the GOUtype process are illustrated showing some of their basic properties as well assome time series realizations. These examples also present their theoreticaland empirical autocorrelation or normalized codifference functions depending onwhether the process has a finite or infinite second moment. We also present themaximum likelihood estimation as well as the Bayesian estimation procedures forthe so-called \emph{Cosine process}, a particular process in the class of GOUtype processes. For the Bayesian estimation method, we consider the powerseries representation of Fox's H-function to better approximate the densityfunction of a random variable $\alpha$-stable distributed. We consider fourgoodness-of-fit tests for helping to decide which \emph{Cosine process} (drivenby a Gaussian or an $\alpha$-stable noise) best fit real data sets. Twoapplications of GOU type model are presented: one based on the Apple companystock market price data and the other based on the cardiovascular mortality inLos Angeles County data.

A Generalization of the Ornstein-Uhlenbeck Process: Theoretical Results, Simulations and Parameter Estimation