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

J. Stein, A. V. Medino, R. M. de Souza and S. R. C. Lopes
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J. Stein: Federal Institute Sul-rio-grandense
A. V. Medino: University of Brasília, Math Department
R. M. de Souza: Federal Technology University of Paraná, Dean’s Office of Research and Graduate Studies
S. R. C. Lopes: Federal University of Rio Grande do Sul, Mathematics and Statistics Institute

A chapter in Time Series and Wavelet Analysis, 2024, pp 111-132 from Springer

Abstract: Abstract 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 α $$\alpha $$ -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 α $$\alpha $$ -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.

Date: 2024
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-031-66398-7_6

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DOI: 10.1007/978-3-031-66398-7_6

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