Stable distributions are a celebrated class of probability laws used in various fields. The $\alpha$-stable process, and its exponentially tempered counterpart, the Classical Tempered Stable (CTS) process, are also prominent examples of L\'evy processes. Simulating these processes is critical for many applications, yet it remains computationally challenging, due to their infinite jump activity. This survey provides an overview of the key properties of these objects offering a roadmap for practitioners. The first part is a review of the stability property, sampling algorithms are provided along with numerical illustrations. Then CTS processes are presented, with the Baeumer-Meerschaert algorithm for increment simulation, and a computational analysis is provided with numerical illustrations across different time scales.

翻译：稳定分布是概率论中一类著名的概率定律，广泛应用于多个领域。$\alpha$-稳定过程及其指数缓释变体——经典缓释稳定（CTS）过程，也是Lévy过程中重要的范例。由于这些过程具有无限跳跃活性，其模拟在众多应用中至关重要，但计算上仍具挑战性。本综述系统梳理了这些对象的关键性质，为实践者提供技术路线图。第一部分回顾稳定性性质，并提供抽样算法及数值示例。随后介绍CTS过程，阐述用于增量模拟的Baeumer-Meerschaert算法，并通过不同时间尺度的数值算例进行计算分析。