A Generalization of the Ornstein-Uhlenbeck Process: Theoretical Results, Simulations and Parameter Estimation
Ornstein–Uhlenbeck 过程的推广：理论、模拟与参数估计AI 解读 ↓
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Ornstein–Uhlenbeck 过程的推广：理论、模拟与参数估计
- 研究问题
- 在更一般的记忆核及多类随机噪声下，推广的 OU 型过程何时具有平稳性和马尔可夫性，又如何估计并比较具体模型？
- 方法与思路
- 作者分析记忆核条件及 Lévy 噪声下的无限可分性，对余弦过程开展似然和贝叶斯估计，并以拟合检验比较噪声设定。
- 材料陈述的贡献
- 摘要称给出平稳性与马尔可夫性的必要充分条件、无限可分表示及过程实例，并展示股票价格和心血管死亡数据应用。
- 阅读关注点
- 应核对记忆核与噪声条件、有限和无限二阶矩下依赖度量的区别，以及稳定分布密度近似对参数估计的影响。
- 阅读边界
- 这是 arXiv 预印本，不能据此确认同行评审或期刊接收；摘要未给出具体条件或拟合结果，股票实例不足以支持定价结论。
原文摘要
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 $α$-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. 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 $α$-stable distributed. We consider four goodness-of-fit tests for helping to decide which \emph{Cosine process} (driven by a Gaussian or an $α$-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.