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Journal of Theoretical Probability

, Volume 19, Issue 2, pp 411–446 | Cite as

Infinite Divisibility for Stochastic Processes and Time Change

  • Ole E. Barndorff-Nielsen
  • Makoto Maejima
  • Ken-iti Sato
Article

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.

Keywords

Infinite divisibility selfdecomposability temporal class Lm property time change chronometer 

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References

  1. 1.
    Akita K., Maejima M. (2002). On certain self-decomposable self-similar processes with independent increments. Statist. Probab. Letters 59, 53–59CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Barndorff-Nielsen O.E., Blæsild P., Schmiegel J. (2004). A parsimonious and universal description of turbulent velocity increments. Eur. Phys. J. B 41, 345–363CrossRefGoogle Scholar
  3. 3.
    Barndorff-Nielsen O.E.., Nicolato E., Shephard N. (2002). Some recent developments in stochastic volatility modelling. Quantitative Finance 2, 11–23CrossRefMathSciNetGoogle Scholar
  4. 4.
    Barndorff-Nielsen O.E.., Pedersen J., Sato K. (2001). Multivariate subordination, selfdecomposability and stability. Adv. Appl. Probab. 33, 160–187CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Barndorff-Nielsen O.E.., Shephard N. (2001). Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics (with Discussion). J. R. Statist. Soc. B 63, 167–241CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Barndorff-Nielsen O.E.., Shephard N. (2006). Impact of jumps on returns and realised variances: econometric analysis of time-deformed L évy processes. J. Econometrics 131, 217–252CrossRefMathSciNetGoogle Scholar
  7. 7.
    Barndorff-Nielsen O. E., and Shephard N. (2006). Continuous Time Approach to Financial Volatility. Cambridge University Press. (To appear.)Google Scholar
  8. 8.
    Barndorff-Nielsen O.E., and Shiryaev A.N. (2007). Change of Time and Change of Measure. (In preparation.)Google Scholar
  9. 9.
    Bertoin J. (1996). Lévy Processes. Cambridge University Press, CambridgeMATHGoogle Scholar
  10. 10.
    Bertoin J. (1997). Subordinators: examples and Applications. In Bernard P. (ed.), Lectures on Probability Theory and Statistics. Ecole d’Eté de Probabilities de Saint-Flour XXVII–1997, pp. 1–91Google Scholar
  11. 11.
    Bochner S. (1949). Diffusion equation and stochastic processes. P. Nat. Acad. Sci. 85, 369–370Google Scholar
  12. 12.
    Bochner S. (1955). Harmonic Analysis and the Theory of Probability. University of California Press, Berkeley and Los AngelesMATHGoogle Scholar
  13. 13.
    Carr P., Geman H., Madan D.B., Yor M. (2002). The fine structure of asset returns: an empirical investigation. J. Business 75, 305–332CrossRefGoogle Scholar
  14. 14.
    Carr P., Geman H., Madan D.B., Yor M. (2003). Stochastic volatility for Lévy processes. Math. Finance 13, 345–382CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Chung K.L., Zambrini J.-C. (2003). Introduction to Random Time and Quantum Randomness. World Scientific Publishing, SingaporeMATHGoogle Scholar
  16. 16.
    Cont R., Tankov P. (2003). Financial Modelling with Jump Processes. Chapman and Hall/CRC Press, LondonGoogle Scholar
  17. 17.
    Eberlein E. (2001). Application of generalized hyperbolic Lévy motions to finance. In: Barndorff-Nielsen O.E., Mikosch T., Resnick S. (eds), Lévy Processes: Theory and Applications. Birkhäuser Verlag, Basel, pp. 319–337Google Scholar
  18. 18.
    Eberlein E., Prause K. (2002). The generalized hyperbolic model: financial derivatives and risk measures. In: Geman H., Madan D., Pliska S., Vorst T. (eds), Mathematical Finance–Bachelier Congress 2000. Springer Verlag, Berlin, pp. 245–267Google Scholar
  19. 19.
    Embrechts P., Maejima M. (2002). Selfsimilar Processes. Princeton Univ Press, PrincetonMATHGoogle Scholar
  20. 20.
    Geman H., Madan D.B., Yor M. (2001). Time changes for Lévy processes. Math. Finance 11, 79–96CrossRefMathSciNetMATHGoogle Scholar
  21. 21.
    Geman H., Madan D.B., Yor M. (2002). Stochastic volatility, jumps and hidden time changes. Finance and Stochastics 6, 63–90CrossRefMathSciNetMATHGoogle Scholar
  22. 22.
    Itô K., McKean H.P., Jr. (1965). Diffusion Processes and Their Sample Paths. Springer, BerlinMATHGoogle Scholar
  23. 23.
    Jeanblanc M., Pitman J., Yor M. (2002). Self-similar processes with independent increments associated with Lévy and Bessel processes. Stoch. Proc. Appl. 100, 223–231CrossRefMathSciNetMATHGoogle Scholar
  24. 24.
    Kasahara Y., Maejima M., Vervaat W. (1988). Log-fractional stable processes. Stoch. Proc. Appl. 30, 329–339CrossRefMathSciNetMATHGoogle Scholar
  25. 25.
    Maejima M., Sato K. (2003). Semi-Lévy processes, semi-selfsimilar additive processes, and semi-stationary Ornstein–Uhlenbeck type processes. J. Math. Kyoto Univ. 43, 609–639MathSciNetMATHGoogle Scholar
  26. 26.
    Maejima M., Sato K., Watanabe T. (2000). Distributions of selfsimilar and semi-selfsimilar processes with independent increments. Statist. Probab. Letters 47, 395–401CrossRefMathSciNetMATHGoogle Scholar
  27. 27.
    Maejima M., Suzuki K., Tamura Y. (1999). Some multivariate infinitely divisible distributions and their projections. Probab. Theory Math. Statist. 19, 421–428MathSciNetMATHGoogle Scholar
  28. 28.
    Maruyama G. (1970). Infinitely divisible processes. Theory Probab. Appl. 15, 1–22CrossRefMathSciNetGoogle Scholar
  29. 29.
    Pedersen J., Sato K. (2003). Cone-parameter convolution semigroups and their subordination. Tokyo J. Math. 26, 503–525MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Rocha-Arteaga A., and Sato K. (2003) Topics in Infinitely Divisible Distributions and Lévy Processes. Aportaciones Matemáticas, Investigación 17, Sociedad Matemática Mexicana.Google Scholar
  31. 31.
    Samorodnitsky G., Taqqu M.S. (1994). Stable Non-Gaussian Random Processes. Chapman & Hall, New YorkMATHGoogle Scholar
  32. 32.
    Sato K. (1991). Self-similar processes with independent increments. Probab. Th. Rel. Fields 89, 285–300CrossRefMATHGoogle Scholar
  33. 33.
    Sato K. (1998). Multivariate distributions with selfdecomposable projections. J. Korean Math. Soc. 35, 783–791MathSciNetMATHGoogle Scholar
  34. 34.
    Sato K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Univ Press, CambridgeMATHGoogle Scholar
  35. 35.
    Sato K. (2004). Stochastic integrals in additive processes and application to semi-Lévy processes. Osaka J. Math. 41, 211–236MathSciNetMATHGoogle Scholar
  36. 36.
    Skorohod A.V. (1991). Random Processes with Independent Increments. Kluwer Academic Pub., Dordrecht, NetherlandsMATHGoogle Scholar
  37. 37.
    Winkel M. (2001). The recovery problem for time-changed Lévy processes. MaPhySto Research Report 2001–37.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • Ole E. Barndorff-Nielsen
    • 1
  • Makoto Maejima
    • 2
  • Ken-iti Sato
    • 3
  1. 1.Department of Mathematical SciencesUniversity of Aarhus, Ny MunkegadeAarhus CDenmark
  2. 2.Department of MathematicsKeio UniversityHiyoshiJapan
  3. 3.Tenpaku-kuJapan

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