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Tempering stable processes. (English) Zbl 1118.60037
Summary: A tempered stable Lévy 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. 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 Lévy 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 Ornstein-Uhlenbeck-type processes.

MSC:
60G52 Stable stochastic processes
60E07 Infinitely divisible distributions; stable distributions
60E10 Characteristic functions; other transforms
60G51 Processes with independent increments; Lévy processes
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