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Infinite Divisibility for Stochastic Processes and Time Change

Author

Listed:
  • Ole E. Barndorff-Nielsen

    (University of Aarhus, Ny Munkegade)

  • Makoto Maejima

    (Keio University)

  • Ken-iti Sato

Abstract

General results concerning infinite divisibility, selfdecomposability, and the class L m property as properties of stochastic processes are presented. A new concept called temporal selfdecomposability of stochastic processes is introduced. Lévy processes, additive processes, selfsimilar processes, and stationary processes of Ornstein–Uhlenbeck type are studied in relation to these concepts. Further, time change of stochastic processes is studied, where chronometers (stochastic processes that serve to change time) and base processes (processes to be time-changed) are independent but do not, in general, have independent increments. Conditions for inheritance of infinite divisibility and selfdecomposability under time change are given.

Suggested Citation

  • Ole E. Barndorff-Nielsen & Makoto Maejima & Ken-iti Sato, 2006. "Infinite Divisibility for Stochastic Processes and Time Change," Journal of Theoretical Probability, Springer, vol. 19(2), pages 411-446, June.
    • Handle: RePEc:spr:jotpro:v:19:y:2006:i:2:d:10.1007_s10959-006-0020-7
    DOI: 10.1007/s10959-006-0020-7
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    References listed on IDEAS

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    1. Akita, Koji & Maejima, Makoto, 2002. "On certain self-decomposable self-similar processes with independent increments," Statistics & Probability Letters, Elsevier, vol. 59(1), pages 53-59, August.
    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, July.
    3. Maejima, Makoto & Sato, Ken-iti & Watanabe, Toshiro, 2000. "Distributions of selfsimilar and semi-selfsimilar processes with independent increments," Statistics & Probability Letters, Elsevier, vol. 47(4), pages 395-401, May.
    4. Barndorff-Nielsen, Ole E. & Shephard, Neil, 2006. "Impact of jumps on returns and realised variances: econometric analysis of time-deformed Levy processes," Journal of Econometrics, Elsevier, vol. 131(1-2), pages 217-252.
    5. O. Barndorff-Nielsen & P. Blæsild & J. Schmiegel, 2004. "A parsimonious and universal description of turbulent velocity increments," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 41(3), pages 345-363, October.
    6. Marc Yor & Dilip B. Madan & Hélyette Geman, 2002. "Stochastic volatility, jumps and hidden time changes," Finance and Stochastics, Springer, vol. 6(1), pages 63-90.
    7. 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.
    8. Ole Barndorff-Nielsen & Elisa Nicolato & Neil Shephard, 2002. "Some recent developments in stochastic volatility modelling," Quantitative Finance, Taylor & Francis Journals, vol. 2(1), pages 11-23.
    9. Hélyette Geman & Dilip B. Madan & Marc Yor, 2001. "Time Changes for Lévy Processes," Mathematical Finance, Wiley Blackwell, vol. 11(1), pages 79-96, January.
    10. Kasahara, Yuji & Maejima, Makoto & Vervaat, Wim, 1988. "Log-fractional stable processes," Stochastic Processes and their Applications, Elsevier, vol. 30(2), pages 329-339, December.
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    Cited by:

    1. Ole E. Barndorff-Nielsen & Orimar Sauri & Benedykt Szozda, 2017. "Selfdecomposable Fields," Journal of Theoretical Probability, Springer, vol. 30(1), pages 233-267, March.
    2. Michele Azzone & Roberto Baviera, 2023. "Is (independent) subordination relevant in option pricing?," Papers 2307.08628, arXiv.org, revised Oct 2023.
    3. Shai Covo, 2011. "Two-Parameter Lévy Processes Along Decreasing Paths," Journal of Theoretical Probability, Springer, vol. 24(1), pages 150-169, March.

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