Tempered Infinitely Divisible Distributions and Processes

In this paper, we construct the new class of tempered infinitely divisible (TID) distributions. Taking into account the tempered stable distribution class, as introduced in the seminal work of Rosiński [Stochastic Process. Appl., 117 (2007), pp. 677–707], a modification of the tempering function allows one to obtain suitable properties. In particular, TID distributions may have exponential moments of any order and conserve all proper properties of the Rosiński setting. Furthermore, we prove that the modified tempered stable distribution is TID and give some further parametric examples.