Stable Lévy process delayed by tempered stable subordinator

Abstract

We consider symmetric stable Lévy motion time-changed by tempered stable subordinator. This process generalizes the normal inverse Gaussian process without drift term, introduced by Barndorff-Nielsen. The asymptotic tail behavior of the density function of this process and corresponding Lévy density is obtained. The governing Fokker–Planck–Kolmogorov equation of the density function of the introduced process in terms of shifted fractional derivative is established. Codifference and asymptotic behavior of the moments are discussed. Further, we also introduce and analyze stable subordinator delayed by tempered stable subordinator.