Tempering stable processes

A tempered stable Levy process combines both the alpha-stable and Gaussian trends. In a short time frame it is close to an alpha-stable process while in a long time frame it approximates a Brownian motion. In this paper we consider a general and robust class of multivariate tempered stable distributions and establish their identifiable parametrization. We prove short and long time behavior of tempered stable Levy processes and investigate their absolute continuity with respect to the underlying alpha-stable processes. We find probabilistic representations of tempered stable processes which specifically show how such processes are obtained by cutting (tempering) jumps of stable processes. These representations exhibit alpha-stable and Gaussian tendencies in tempered stable processes and thus give probabilistic intuition for their study. Such representations can also be used for simulation. We also develop the corresponding representations for Omstein-Uhlenbeck-type processes. (C) 2006 Elsevier B.V. All rights reserved.