Inverse tempered stable subordinators

We consider the first-exit time of a tempered ββ-stable subordinator, also called inverse tempered stable (ITS) subordinator. An integral form representation and a series representation for the density of the ITS subordinator are obtained. The asymptotic behavior of the qq-th order moments of the ITS subordinator are investigated. The limiting form of the ITS density and its kk-th order derivatives are derived as the space variable x→0+x→0+. Finally, the governing pde of the ITS density is also obtained.