Stable and tempered stable distributions and processes: an overview toward trajectory simulation

TL;DR

This paper reviews stable and tempered stable distributions and processes, focusing on their properties and simulation methods, including algorithms and numerical illustrations, to aid practitioners in modeling complex stochastic systems.

Contribution

It provides a comprehensive overview of stable and tempered stable processes, including new sampling algorithms and computational analyses for simulating these complex Lévy processes.

Findings

Sampling algorithms for stable distributions are detailed.

Numerical illustrations demonstrate the effectiveness of simulation methods.

Analysis across different time scales enhances understanding of process behavior.

Abstract

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.