Fast simulation of tempered stable Ornstein–Uhlenbeck processes

Abstract

Constructing Lévy-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 a-remainder of their self-decomposable stationary laws. In the present study we fully characterize the Lévy triplet of these $$a$$ a -remainders 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 procedures which are tens of times faster than those already available in the existing literature.

Efficient Simulation of $p$-Tempered $α$-Stable OU Processes

“…We now show that the IGa distribution is a GGSM. When p = 1 and γ = 1, this was already observed in [28].…”

“…These can then be used to simulate the corresponding OU process on a finite grid. Our simulation methods extend the ideas introduced in [28] for CTS distributions and work well in practice. The main idea is based on showing that certain components of the transition law can be represented as generalized gamma scale mixtures (GGSMs).…”

“…Simulation from this distribution was studied in [22] and [28]. We follow the approach given in [28].…”

“…The second, denoted OUTS processes, correspond to the case where the background driving Lévy process (BDLP) has a TS distribution. The study of the transition laws of TSOU processes has been primarily focused on the fairly simple univariate class of so-called classical tempered stable (CTS) distributions, see [35], [32], [19], [22], [28], and the references therein. More detailed results for the special cases of gamma and inverse Gaussian distributions are given in [34], [21], and [27].…”

“…Significantly less attention has been paid to the class of OUTS processes. To the best of our knowledge, in this case, the transition laws have only been studied in the case of gamma distributions in [21] and [27], and in the case of CTS distributions with parameter α ∈ [0, 1) in [22] and [28]. Although, it should be noted that some preliminary results about certain univariate TS distributions beyond CTS are given in [22].…”