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On finite truncation of infinite shot noise series representation of tempered stable laws

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  • Imai, Junichi
  • Kawai, Reiichiro

Abstract

Tempered stable processes are widely used in various fields of application as alternatives with finite second moment and long-range Gaussian behaviors to stable processes. Infinite shot noise series representation is the only exact simulation method for the tempered stable process and has recently attracted attention for simulation use with ever improved computational speed. In this paper, we derive series representations for the tempered stable laws of increasing practical interest through the thinning, rejection, and inverse Lévy measure methods. We make a rigorous comparison among those representations, including the existing one due to Imai and Kawai [29] and Rosiński (2007) [3], in terms of the tail mass of Lévy measures which can be simulated under a common finite truncation scheme. The tail mass are derived in closed form for some representations thanks to various structural properties of the tempered stable laws. We prove that the representation via the inverse Lévy measure method achieves a much faster convergence in truncation to the infinite sum than all the other representations. Numerical results are presented to support our theoretical analysis.

Suggested Citation

  • Imai, Junichi & Kawai, Reiichiro, 2011. "On finite truncation of infinite shot noise series representation of tempered stable laws," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(23), pages 4411-4425.
  • Handle: RePEc:eee:phsmap:v:390:y:2011:i:23:p:4411-4425
    DOI: 10.1016/j.physa.2011.07.028
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    References listed on IDEAS

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    1. 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, July.
    2. Matsushita, Raul & Rathie, Pushpa & Da Silva, Sergio, 2003. "Exponentially damped Lévy flights," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 326(3), pages 544-555.
    3. Heyde, C.C. & Sly, Allan, 2008. "A Cautionary note on modeling with fractional Lévy flights," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(21), pages 5024-5032.
    4. Fred Espen Benth & Martin Groth & Rodwell Kufakunesu, 2007. "Valuing Volatility and Variance Swaps for a Non-Gaussian Ornstein-Uhlenbeck Stochastic Volatility Model," Applied Mathematical Finance, Taylor & Francis Journals, vol. 14(4), pages 347-363.
    5. Vinogradov, Dmitry V., 2010. "Cumulant approach of arbitrary truncated Levy flight," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(24), pages 5794-5800.
    6. Kawai Reiichiro, 2006. "An importance sampling method based on the density transformation of Lévy processes," Monte Carlo Methods and Applications, De Gruyter, vol. 12(2), pages 171-186, April.
    7. Karen J. Palmer & Martin S. Ridout & Byron J. T. Morgan, 2008. "Modelling cell generation times by using the tempered stable distribution," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 57(4), pages 379-397, September.
    8. Houdré, C. & Kawai, R., 2006. "On fractional tempered stable motion," Stochastic Processes and their Applications, Elsevier, vol. 116(8), pages 1161-1184, August.
    9. Gupta, Hari M. & Campanha, José R., 1999. "The gradually truncated Lévy flight for systems with power-law distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 268(1), pages 231-239.
    10. 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.
    11. Dmitry V. Vinogradov, 2010. "Cumulant Approach of Arbitrary Truncated Levy Flight," Papers 1006.2489, arXiv.org, revised Oct 2010.
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    Cited by:

    1. Kawai, Reiichiro, 2021. "A general approach to sample path generation of infinitely divisible processes via shot noise representation," Statistics & Probability Letters, Elsevier, vol. 174(C).
    2. Michele Leonardo Bianchi & Svetlozar T. Rachev & Frank J. Fabozzi, 2018. "Calibrating the Italian Smile with Time-Varying Volatility and Heavy-Tailed Models," Computational Economics, Springer;Society for Computational Economics, vol. 51(3), pages 339-378, March.
    3. Hasan Fallahgoul & Gregoire Loeper, 2021. "Modelling tail risk with tempered stable distributions: an overview," Annals of Operations Research, Springer, vol. 299(1), pages 1253-1280, April.
    4. Reiichiro Kawai, 2017. "Sample Path Generation of Lévy-Driven Continuous-Time Autoregressive Moving Average Processes," Methodology and Computing in Applied Probability, Springer, vol. 19(1), pages 175-211, March.
    5. Till Massing, 2018. "Simulation of Student–Lévy processes using series representations," Computational Statistics, Springer, vol. 33(4), pages 1649-1685, December.
    6. Michele Bianchi & Frank Fabozzi, 2014. "Discussion of ‘on simulation and properties of the stable law’ by Devroye and James," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 23(3), pages 353-357, August.

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