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Tempering stable processes

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  • Rosinski, Jan

Abstract

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.

Suggested Citation

  • Rosinski, Jan, 2007. "Tempering stable processes," Stochastic Processes and their Applications, Elsevier, vol. 117(6), pages 677-707, June.
    • Handle: RePEc:eee:spapps:v:117:y:2007:i:6:p:677-707
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    References listed on IDEAS

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    2. Sztonyk, Pawel, 2011. "Transition density estimates for jump Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 121(6), pages 1245-1265, June.
    3. Jim E. Griffin & Fabrizio Leisen, 2017. "Compound random measures and their use in Bayesian non-parametrics," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(2), pages 525-545, March.
    4. Sebastian, Orzeł & Agnieszka, Wyłomańska, 2010. "Calibration of the subdiffusive arithmetic Brownian motion with tempered stable waiting-times," MPRA Paper 28593, University Library of Munich, Germany.
    5. Gong, Xiaoli & Zhuang, Xintian, 2017. "American option valuation under time changed tempered stable Lévy processes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 466(C), pages 57-68.
    6. Choi, Jaehyung & Kim, Young Shin & Mitov, Ivan, 2015. "Reward-risk momentum strategies using classical tempered stable distribution," Journal of Banking & Finance, Elsevier, vol. 58(C), pages 194-213.
    7. Szymon Borak & Adam Misiorek & Rafał Weron, 2010. "Models for Heavy-tailed Asset Returns," SFB 649 Discussion Papers SFB649DP2010-049, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    8. Svetlana Boyarchenko & Sergei Levendorskiĭ, 2019. "Sinh-Acceleration: Efficient Evaluation Of Probability Distributions, Option Pricing, And Monte Carlo Simulations," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(03), pages 1-49, May.
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    10. Gong, Xiaoli & Zhuang, Xintian, 2016. "Option pricing for stochastic volatility model with infinite activity Lévy jumps," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 455(C), pages 1-10.
    11. Hounyo, Ulrich & Varneskov, Rasmus T., 2017. "A local stable bootstrap for power variations of pure-jump semimartingales and activity index estimation," Journal of Econometrics, Elsevier, vol. 198(1), pages 10-28.
    12. Gong, Xiaoli & Zhuang, Xintian, 2017. "Pricing foreign equity option under stochastic volatility tempered stable Lévy processes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 483(C), pages 83-93.
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