Fast simulation of tempered stable Ornstein-Uhlenbeck processes

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 a-remainder of their self-decomposable stationary laws. In the present study we fully characterize the Levy triplet of these 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.