The tempered stable process with infinitely divisible inverse subordinators

Abstract

In the last decade processes driven by inverse subordinators have become extremely popular. They have been used in many different applications, especially for data with observable constant time periods. However, the classical model, i.e. the subordinated Brownian motion, can be inappropriate for the description of observed phenomena that exhibit behavior not adequate for Gaussian systems. Therefore, in this paper we extend the classical approach and replace the Brownian motion by the tempered stable process. Moreover, on the other hand, as an extension of the classical model, we analyze the general class of inverse subordinators. We examine the main properties of the tempered stable process driven by inverse subordinators from the infinitely divisible class of distributions. We show the fractional Fokker–Planck equation of the examined process and the asymptotic behavior of the mean square displacement for two cases of subordinators. Additionally, we examine how an external force can influence the examined characteristics.