Time Change, Volatility, and Turbulence

  • Ole E. Barndorff-Nielsen
  • Jürgen Schmiegel


A concept of volatility modulated Volterra processes is introduced. Apart from some brief discussion of generalities, the paper focusses on the special case of backward moving average processes driven by Brownian motion. In this framework, a review is given of some recent modelling of turbulent velocities and associated questions of time change and universality. A discussion of similarities and differences to the dynamics of financial price processes is included.


Time Change Fractional Brownian Motion Inertial Range Stochastic Volatility Modelling Velocity Increment 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Andersen TG, Bollerslev T, Diebold FX, Ebens H (2001) The distribution of realized stock return volatility. J. Fin. Econometrics, 61:43–76.CrossRefGoogle Scholar
  2. 2.
    . Andersen TG, Bollerslev T, Frederiksen PH, Nielsen MØ (2006) Continuous time models, realized volatilities, and testable distributional implications for daily stock returns. Unpublished paper.Google Scholar
  3. 3.
    Asmussen S, Rosinski J (2001) Approximation of small jumps of Lévy processes with a view towards simulation. J. Appl. Probab., 38:482–493.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Barndorff-Nielsen OE (1977) Exponentially decreasing distributions for the log arithm of particle size. Proc. R. Soc. London A 353:401–419.CrossRefGoogle Scholar
  5. 5.
    . Barndorff-Nielsen OE (1995) Normal inverse Gaussian processes and the modelling of stock returns. Research Report 300, Dept. Theor. Statistics, Aarhus University.Google Scholar
  6. 6.
    Barndorff-Nielsen OE (1997) Normal inverse Gaussian distributions and stochastic volatility modelling. Scand. J. Statist., 24:1–14.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Barndorff-Nielsen OE (1998) Probability and statistics: self-decomposability, finance and turbulence. In: Acccardi L, Heyde CC (Eds) Probability Towards 2000,47–57. Proceedings of a Symposium held 2–5 October 1995 at Columbia University. Springer-Verlag, New York.Google Scholar
  8. 8.
    Barndorff-Nielsen OE (1998) Processe of normal inverse Gaussian type. Finance and Stochastics, 2:41–68.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Barndorff-Nielsen OE (1998) Superposition of Ornstein-Uhlenbeck type pro cesses. Theory Prob. Its Appl., 45:175–194.CrossRefMathSciNetGoogle Scholar
  10. 10.
    Barndorff-Nielsen OE, Blæsild P, Schmiegel J (2004) A parsimonious and uni versal description of turbulent velocity increments. Eur. Phys. J., B 41:345–363. Time Change, Volatility, and Turbulence 49.Google Scholar
  11. 11.
    Barndorff-Nielsen OE, Graversen SE, Jacod J, Podolskij M, Shephard N (2006) A central limit theorem for realised power and bipower variations of continuous semimartingales. In: Kabanov Yu, Liptser R, Stoyanov J (Eds) From Stochastic Calculus to Mathematical Finance 33–68. Festschrift in Honour of A.N. Shiryaev. Springer, Heidelberg.CrossRefGoogle Scholar
  12. 12.
    Barndorff-Nielsen OE, Graversen SE, Jacod J, Shephard N (2006) Limit the orems for bipower variation in financial econometrics. Econometric Theory, 22:677–719.zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Barndorff-Nielsen OE, Levendorski˘ı SZ (2001) Feller processes of normal inverse Gaussian type. Quantitative Finance, 1:318–331.CrossRefMathSciNetGoogle Scholar
  14. 14.
    Barndorff-Nielsen OE, Prause K (2001) Apparent scaling. Finance and Stochastics, 5:103–113.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Barndorff-Nielsen OE, Schmiegel J (2004) Lévy-based tempo-spatial modelling; with applications to turbulence. Uspekhi Mat. Nauk., 59:65–91.MathSciNetGoogle Scholar
  16. 16.
    . Barndorff-Nielsen OE, Schmiegel J (2006) A stochastic differential equation framework for the timewise dynamics of turbulent velocities. Theory of Probability and its Applications. (To appear).Google Scholar
  17. 17.
    . Barndorff-Nielsen OE, Schmiegel J (2006) Time change and universality in turbulence. (Submitted).Google Scholar
  18. 18.
    Barndorff-Nielsen OE, Schmiegel J. (2007) Ambit processes; with applications to turbulence and cancer growth. Proceedings of the 2005 Abel Symposium on Stochastic Analysis and Applications. Springer, Heidelberg (To appear).Google Scholar
  19. 19.
    . Barndorff-Nielsen OE, Schmiegel J (2007) Change of time and universal laws in Turbulence. (Submitted).Google Scholar
  20. 20.
    . Barndorff-Nielsen OE, Schmiegel J, Shephard N (2006) Time change and universality in turbulence and finance. (Submitted).Google Scholar
  21. 21.
    . Barndorff-Nielsen OE, Schmiegel J, Shephard N (2007) QV, BV and VR under stationary Gaussian processes. (In preparation).Google Scholar
  22. 22.
    Barndorff-Nielsen OE, Shephard N (2001) Modelling by Lévy processes for financial econometrics. In: Barndorff-Nielsen OE, Mikosch T, Resnick S (Eds) Lévy Processes - Theory and Applications, 283–318. Birkhäuser, Boston.Google Scholar
  23. 23.
    Barndorff-Nielsen OE, Shephard N (2001) Non-Gaussian Ornstein-Uhlenbeck based models and some of their uses in financial economics (with Discussion). J. R. Statist. Soc., B 63:167–241.CrossRefMathSciNetGoogle Scholar
  24. 24.
    Barndorff-Nielsen OE, Shephard N (2002) Integrated OU processes and non Gaussian OU-based stochastic volatility. Scand. J. Statist., 30:277–295.CrossRefMathSciNetGoogle Scholar
  25. 25.
    Barndorff-Nielsen OE, Shephard N (2004) Power and bipower variation with stochastic volatility and jumps (with Discussion). J. Fin. Econometrics, 2:1–48.CrossRefGoogle Scholar
  26. 26.
    . Barndorff-Nielsen OE, Shephard N (2008) Financial Volatility in Continous Time. Cambridge University Press. (To appear).Google Scholar
  27. 27.
    Barndorff-Nielsen OE, Shiryaev AN (2008) Change of Time and Change of Mea sure. World Scientific, Singapore (To appear).Google Scholar
  28. 28.
    . Bender C, Marquardt T (2007) Stochastic calculus for convoluted Lévy processes. (Unpublished manuscript).Google Scholar
  29. 29.
    Castaing B, Gagne Y, Hopfinger EJ (1990) Velocity probability density functions of high Reynolds number turbulence. Physica, D 46:177–200.zbMATHCrossRefGoogle Scholar
  30. 30.
    Cont R, Tankov P (2004) Financial Modelling With Jump Processes. Chapman & Hall/CRC, London.zbMATHGoogle Scholar
  31. 31.
    Decreusefond L (2005) Stochastic integration with respect to Volterra processes. Ann. I. H. Poincaré, PR41:123–149. 50 Ole E. Barndorff-Nielsen and Jürgen Schmiegel.CrossRefMathSciNetGoogle Scholar
  32. 32.
    Decreusefond L, Savy N (2006) Anticipative calculus with respect to filtered Poisson processes. Ann. I. H. Poincaré, PR42:343–372.CrossRefMathSciNetGoogle Scholar
  33. 33.
    Eberlein E (2000) Application of generalized hyperbolic Lévy motion to finance. In: Barndorff-Nielsen OE, Mikosch T, Resnick S (Eds) Lévy Processes - Theory and Applications, 319–336. Birkhäuser, Boston.Google Scholar
  34. 34.
    Elsner JW, Elsner W (1996) On the measurement of turbulence energy dissipation. Meas. Sci. Technol., 7:1334–1348.CrossRefGoogle Scholar
  35. 35.
    . Forsberg L (2002) On the Normal Inverse Gaussian distribution in Modelling Volatility in the Financial Markets. Acta Universitatis Upsaliemsis, Studia Statistica Upsaliensia 5, Uppsala.Google Scholar
  36. 36.
    Ghashgaie S, Breymann W, Peinke J, Talkner P, Dodge Y (1996) Turbulent cascades in foreign exchange markets. Nature 381:767–770.CrossRefGoogle Scholar
  37. 37.
    Hult H (2003) Approximating some Volterra type stochastic intergrals with applications to parameter estimation. Stoch. Proc. Appl., 105:1–32.zbMATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Kolmogorov AN (1941) Dissipation of energy in locally isotropic turbulence. Dokl. Akad. Nauk. SSSR, 32:16–18.Google Scholar
  39. 39.
    Kolmogorov AN (1962) A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. J. Fluid Mech, 13:82–85.zbMATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Marquardt T (2006) Fractional Lévy processes with an application to long memory moving average processes. Bernoulli, 12:1099–1126.zbMATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    McNeil AJ, Frey R, Embrechts P (2005) Quantitative Risk Management. Princeton University Press, Princeton.zbMATHGoogle Scholar
  42. 42.
    Norros I, Valkeila E, Virtamo J (1999) An elementary approach to a Girsanov formula and other analytic results on fractional Brownian motion. Bernoulli, 5:571–587.zbMATHCrossRefMathSciNetGoogle Scholar
  43. 43.
    Obukhov AM (1962) Some specific features of atmospheric turbulence. J. Fluid Mech., 13:77–81.CrossRefMathSciNetGoogle Scholar
  44. 44.
    Peinke J, Bottcher F, Barth S (2004) Anomalous statistics in turbulence, financial markets and other complex systems. Ann. Phys. (Leipzig), 13:450–460.zbMATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    Prause K (1999) The Generalized Hyperbolic Model: Estimation, Financial Derivatives and Risk Measures. Dissertation, Albert-Ludwigs-Universität, Freiburg i. Br.zbMATHGoogle Scholar
  46. 46.
    Raible S (2000) Lévy Processes in Finance: Theory, Numerics, Empirical Facts. Dissertation, Albert-Ludwigs-Universität, Freiburg i. Br.Google Scholar
  47. 47.
    Rydberg TH (1997) The normal inverse Gaussian Lévy process: simulation and approximation. Comm. Statist.: Stochastic Models, 13:887–910.zbMATHCrossRefMathSciNetGoogle Scholar
  48. 48.
    Rydberg TH (1999) Generalized hyperbolic diffusions with applications towards finance. Math. Finance, 9:183–201.zbMATHCrossRefMathSciNetGoogle Scholar
  49. 49.
    Samorodnitsky G, Taqqu MS (1994) Stable Non-Gaussian Random Processes. Chapman and Hall, New York.zbMATHGoogle Scholar
  50. 50.
    . Shiryaev AN (2006) Kolmogorov and the turbulence. Research Report 2006–4. Thiele Centre for Applied Mathematics in Natural Science.Google Scholar
  51. 51.
    . Shiryaev AN (2007) On the classical, statistical and stochastic approaches to hydrodynamic turbulence. Research Report 2007–2. Thiele Centre for Applied Mathematics in Natural Science.Google Scholar
  52. 52.
    Vincent A, Meneguzzi M (1991) The spatial structure and statistical properties of homogeneous turbulence. J. Fluid Mech., 225:1–25.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Ole E. Barndorff-Nielsen
    • 1
  • Jürgen Schmiegel
    • 1
  1. 1.The T.N. Thiele Centre for Applied Mathematics in Natural Science, Department of Mathematical SciencesUniversity of AarhusAarhus CDenmark

Personalised recommendations