Skip to main content

Probability and Statistics: Self-Decomposability, Finance and Turbulence

  • Chapter
Probability Towards 2000

Part of the book series: Lecture Notes in Statistics ((LNS,volume 128))

Abstract

After some general remarks about the relationship between probability and statistics, a discussion is given of closely similar, key features of empirical data from finance and from turbulence, and this is followed by an account of recent work on stochastic modelling incorporating those features.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Andersen, P.K., Borgan, Ø., Gill, R.D. and Keiding, N. (1993): Statistical Models based on Counting Processes. Heidelberg: Springer-Verlag.

    MATH  Google Scholar 

  2. Barndorff-Nielsen, O.E. (1977): Exponentially decreasing distributions for the logarithm of particle size. Proc. R. Soc. London A 353, 401–419.

    Google Scholar 

  3. Barndorff-Nielsen, O.E. (1978): Hyperbolic distributions and distributions on hyperbolae. Scand. J. Statist. 5, 151–157.

    MathSciNet  MATH  Google Scholar 

  4. Barndorff-Nielsen, O.E. (1979): Models for non-Gaussian variation; with applications to turbulence. Proc. R. Soc. London A 368, 501–520.

    MathSciNet  Google Scholar 

  5. Barndorff-Nielsen, O.E. (1986): Sand, wind and statistics. Acta Mechanica 64, 1–18.

    Article  Google Scholar 

  6. Barndorff-Nielsen, O.E. (1993): Important areas for future statistical research. Statist Comp. 3, 181.

    Article  Google Scholar 

  7. Barndorff-Nielsen, O.E. (1995): Normal inverse Gaussian processes and the modelling of stock returns. Research Report 300. Dept. Theor. Statistics, Aarhus University.

    Google Scholar 

  8. Barndorff-Nielsen, O.E. (1996a): Normal inverse Gaussian distributions and stochastic volatility modelling. Scand. J. Statist. (To appear.)

    Google Scholar 

  9. Barndorff-Nielsen, O.E.(1996b): Processes of normal inverse Gaussian type. Finance and Stochastics. (To appear.)

    Google Scholar 

  10. Barndorff-Nielsen, O.E., Jensen, J.L. and Sørensen, M. (1989): Wind shear and hyperbolic distributions. Boundary-Layer Meteorology 46, 417–431.

    Article  Google Scholar 

  11. Barndorff-Nielsen, O.E., Jensen, J.L. and Sørensen, M. (1990): Parametric modelling of turbulence. Phil Trans. R. Soc. London A 332, 439–455.

    Article  Google Scholar 

  12. Barndorff-Nielsen, O.E., Jensen, J.L. and Sørensen, M. (1993): A statistical model for the streamwise component of a turbulent wind field. Ann. Geophysicae 11, 99–103.

    Google Scholar 

  13. Barndorff-Nielsen, O.E., Jensen, J.L. and Sørensen, M. (1995): Some stationary processes in discrete and continuous time. Research Report 241. Dept. Theor. Statist., Aarhus University.

    Google Scholar 

  14. Barndorff-Nielsen, O.E., Kent, J. and Sørensen, M. (1982): Normal variance mean mixtures and the z-distributions. Int. Statist. Review 50, 145–159.

    Article  MATH  Google Scholar 

  15. Clark, P.K. (1973): A subordinated stochastic process model with finite variance for speculative prices. Econometrica 41, 135–159.

    Article  MathSciNet  MATH  Google Scholar 

  16. Damien, P., Laud, P.W. and Smith, A.F.M. (1995): Approximate random variable generation from infinitely distributions with applications to Bayesian inference. J. R. Statist Soc. B 57, 547–563.

    MathSciNet  Google Scholar 

  17. Eberlein, E. and Keller, U. (1995): Hyperbolic distributions in finance. Bernoulli 1, 281–299.

    Article  MATH  Google Scholar 

  18. Frisch, U. (1995): Turbulence. The legacy of A.N. Kolmogorov. Cambridge University Press.

    MATH  Google Scholar 

  19. Halgreen, C. (1979): Self-decomposability of the generalized inverse Gaussian and hyperbolic distributions. Z. Wahrscheinlichkeitstheorie verw. Gebiete 47, 13–17.

    Article  MathSciNet  MATH  Google Scholar 

  20. Hassaïri, A. (1992): La classification des families exponentielles naturelles sur Rd par l’action du groupe linéaire de Rd+1. C. R. Acad. Sc. Paris 315, 207–210.

    MATH  Google Scholar 

  21. Hassaïri, A. (1993): Les (d + 3)G-orbites de la classe de Morris-Mora des families exponentielles de R d . C. R. Acad. Sc. Paris 317, 887–890.

    MATH  Google Scholar 

  22. Küchler, U., Neumann, K, Sørensen, M. and Streller, A. (1994): Stock returns and hyperbolic distributions. Discussion Paper No. 23, Sonderforschungsbereich 373, Humboldt Universität zu Berlin.

    Google Scholar 

  23. Letac, G. and Seshadri, V. (1995): A random continued fraction in Rd+l with an inverse Gaussian distribution. Bernoulli 1, 381–393.

    MathSciNet  MATH  Google Scholar 

  24. Mandelbrot, B. and Taylor, H.M. (1967): On the distribution of stock price differences. Operations Research 15, 1057–1062.

    Article  Google Scholar 

  25. Rachev, S.T. and Ruschendorf, L. (1994): Models for option prices. Theory Probab. Appl. 39, 120–152.

    Article  MathSciNet  Google Scholar 

  26. Ripley, B.D. (1988): Statistical Inference for Spatial Processes. Cambridge University Press.

    Google Scholar 

  27. Rydberg, T.H. (1996a): The normal inverse Gaussian Lévy process: approximation and simulation. Revised version to appear in Stochastic Models.

    Google Scholar 

  28. Rydberg, T.H. (1996b): Generalized hyperbolic diffusions with applications towards finance. Submitted to Math. Finance.

    Google Scholar 

  29. Rydberg, T.H. (1996c): Existence of unique equivalent martingale measures in a Markovian setting. Finance and Stochastics. (To appear.)

    Google Scholar 

  30. Shephard, N. (1995): Statistical aspects of ARCH and stochastic volatility. In D.R. Cox, D.V. Hinkley and O.E. Barndorff-Nielsen (Eds.): Time Series Models. In econometrics, finance and other fields. London: Chapman and Hall. Pp. 1–67.

    Google Scholar 

  31. Van Atta, C.W. and Park, J. (1972): Statistical self-similarity and inertial subrange turbulence. In M. Rosenblatt and C. W. van Atta (Eds.): Statistical Models and Turbulence. Lecture Notes in Physics 12. Springer-Verlag: New York. Pp. 402–426.

    Google Scholar 

  32. Winkler, G. (1995): Image Analysis, Random Fields and Dynamic Monte Carlo Methods. A mathematical introduction. Berlin: Springer-Verlag.

    MATH  Google Scholar 

  33. Wyngaard, J.C. and Tennekes, H. (1970): Measurements of the small scale structure of turbulence at moderate Reynolds numbers. Phys. Fluids 13, 1962–1969.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1998 Springer-Verlag New York, Inc.

About this chapter

Cite this chapter

Barndorff-Nielsen, O.E. (1998). Probability and Statistics: Self-Decomposability, Finance and Turbulence. In: Accardi, L., Heyde, C.C. (eds) Probability Towards 2000. Lecture Notes in Statistics, vol 128. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-2224-8_3

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-2224-8_3

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-0-387-98458-2

  • Online ISBN: 978-1-4612-2224-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics