There are two types of tempered stable (TS) based Ornstein–Uhlenbeck (OU) processes: (i) the OU-TS process, the OU process driven by a TS subordinator, and (ii) the TS-OU process, the OU process with TS marginal law. They have various applications in financial engineering and econometrics. In the literature, only the second type under the stationary assumption has an exact simulation algorithm. In this paper we develop a unified approach to exactly simulate both types without the stationary assumption. It is mainly based on the distributional decomposition of stochastic processes with the aid of an acceptance–rejection scheme. As the inverse Gaussian distribution is an important special case of TS distribution, we also provide tailored algorithms for the corresponding OU processes. Numerical experiments and tests are reported to demonstrate the accuracy and effectiveness of our algorithms, and some further extensions are also discussed.

中文翻译：

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Ornstein-Uhlenbeck 回火稳定过程的精确模拟

有两种基于回火稳定 (TS) 的 Ornstein-Uhlenbeck (OU) 过程：(i)OUT-TS流程，由 TS 下属驱动的 OU 进程，以及 (ii)TS-OU流程, 具有 TS 边际律的 OU 过程。它们在金融工程和计量经济学中有各种应用。在文献中，只有静止假设下的第二种类型具有精确的模拟算法。在本文中，我们开发了一种统一的方法来精确模拟这两种类型，而无需静止假设。它主要基于在接受-拒绝方案的帮助下对随机过程的分布分解。由于逆高斯分布是TS分布的一个重要特例，我们还为相应的OU过程提供量身定制的算法。报告了数值实验和测试以证明我们算法的准确性和有效性，并且还讨论了一些进一步的扩展。