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

In this work, we study the class of stochastic process that generalizes the Ornstein-Uhlenbeck processes, hereafter called by 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 Lévy process, a Lévy 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 Lévy noise we prove that it is infinitely divisible showing its generating triplet. We also present the maximum likelihood estimation as well as the Bayesian estimation procedures for the so-called 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 (see Supplementary Material for more details). An application based on the Apple company stock market price data is presented showing the usefulness of the Cosine process.