Tempered stable distributions and finite variation Ornstein-Uhlenbeck processes

TL;DR

This paper characterizes the Le9vy triplet of a-reminders of self-decomposable laws and develops efficient algorithms for simulating Ornstein-Uhlenbeck processes driven by tempered stable distributions, enhancing computational methods.

Contribution

It provides a complete characterization of the Le9vy triplet for a-reminders and introduces a new, more efficient algorithm for simulating Ornstein-Uhlenbeck processes with tempered stable laws.

Findings

Explicit Le9vy triplet characterization for a-reminders.

Development of an exact, computationally efficient simulation algorithm.

Application to exponentially-modulated tempered stable laws.

Abstract

Constructing \Levy-driven Ornstein-Uhlenbeck processes is a task closely related to the notion of self-decomposability. In particular, their transition laws are linked to the properties of what will be hereafter called the \emph{a-reminder} of their self-decomposable stationary laws. In the present study we fully characterize the L\'evy triplet of these a-reminder s and we provide a general framework to deduce the transition laws of the finite variation Ornstein-Uhlenbeck processes associated with tempered stable distributions. We focus finally on the subclass of the exponentially-modulated tempered stable laws and we derive the algorithms for an exact generation of the skeleton of Ornstein-Uhlenbeck processes related to such distributions, with the further advantage of adopting a procedure computationally more efficient than those already available in the existing literature.