  The paper describes the self-decomposable distribution and the background
driving L\'evy process (BDLP) associated with the Generalized Tempered Stable
(GTS) distribution. Two distributions are provided: the background driving
L\'evy process (BDLP) of the GTS distribution and the self-decomposable
distribution generated by the GTS distribution as BDLP. The derived
self-decomposable distribution and the GTS distribution are used as stationary
distribution in the Ornstein-Uhlenbeck type process. A simulation method, based
on sampling the random integral representation, is applied to mimic S&P 500
Index and Bitcoin daily cumulative return process.
