Tempered stable processes

We will discuss the class of tempered stable distributions and processes. Unlike stable distributions, the tempered ones have all moments finite, including exponential moments of some order. Tempered stable Levy processes behave like stable processes in a short period of time while in a long time frame they are approximately Gaussian. Special cases of tempered stable distributions were introduced in statistical physics to model phenomena exhibiting local spatiotemporal fractality and global aggregational Gaussianity, such as turbulence. Certain tempered stable processes appeared also in financial mathematics.