Self-Decomposable Laws Associated with General Tempered Stable (GTS) Distribution and their Simulation Applications

Abstract: 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.