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Tempering stable processes

  • Rosinski, Jan
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    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. 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 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.

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    File URL: http://www.sciencedirect.com/science/article/B6V1B-4M8763W-3/2/9b05860adb7275ee94fc996d3e58374d
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    Article provided by Elsevier in its journal Stochastic Processes and their Applications.

    Volume (Year): 117 (2007)
    Issue (Month): 6 (June)
    Pages: 677-707

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    Handle: RePEc:eee:spapps:v:117:y:2007:i:6:p:677-707
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    1. Ole E. Barndorff-Nielsen & Neil Shephard, 2001. "Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 63(2), pages 167-241.
    2. Peter Carr & Hélyette Geman & Dilip B. Madan & Marc Yor, 2003. "Stochastic Volatility for Lévy Processes," Mathematical Finance, Wiley Blackwell, vol. 13(3), pages 345-382.
    3. Brockett, Patrick L. & Tucker, Howard G., 1977. "A conditional dichotomy theorem for stochastic processes with independent increments," Journal of Multivariate Analysis, Elsevier, vol. 7(1), pages 13-27, March.
    4. Peter Carr & Helyette Geman, 2002. "The Fine Structure of Asset Returns: An Empirical Investigation," The Journal of Business, University of Chicago Press, vol. 75(2), pages 305-332, April.
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