Fractional Normal Inverse Gaussian Process

  • Arun Kumar
  • Palaniappan Vellaisamy


Normal inverse Gaussian (NIG) process was introduced by Barndorff-Nielsen (Scand J Statist 24:1–13, 1997) by subordinating Brownian motion with drift to an inverse Gaussian process. Increments of NIG process are independent and are stationary. In this paper, we introduce dependence between the increments of NIG process, by subordinating fractional Brownian motion to an inverse Gaussian process and call it fractional normal inverse Gaussian (FNIG) process. The basic properties of this process are discussed. Its marginal distributions are scale mixtures of normal laws, infinitely divisible for the Hurst parameter 1/2 ≤ H < 1 and are heavy tailed. First order increments of the process are stationary and possess long-range dependence (LRD) property. It is shown that they have persistence of signs LRD property also. A generalization to an n-FNIG process is also discussed, which allows Hurst parameter H in the interval (n − 1, n). Possible applications to mathematical finance and hydraulics are also pointed out.


Fractional Brownian motion Fractional normal inverse Gaussian process Generalized gamma convolutions Infinite divisibility Long-range dependence Subordination 

AMS 2000 Subject Classifications

Primary 60G22; Secondary 60G07 60G15 


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  1. Applebaum D (2004) Levy processes and stochastic calculus. Cambridge University Press, CambridgeMATHCrossRefGoogle Scholar
  2. Barndorff-Nielsen OE (1997) Normal inverse Gaussian distributions and stochastic volatility modeling. Scand J Statist 24:1–13MathSciNetMATHCrossRefGoogle Scholar
  3. Beran J (1994) Statistics for long-memory processes. Chapman & Hall, New YorkMATHGoogle Scholar
  4. Bertoin J (1996) Levy processes. Cambridge University Press, CambridgeMATHGoogle Scholar
  5. Bondesson L (1979) A general result on infinite divisibility. Ann Probab 7(6):965–979MathSciNetMATHCrossRefGoogle Scholar
  6. Clark PK (1973) A subordinated process model with finite variance for speculative prices. Econometrica 41:135–155MathSciNetMATHCrossRefGoogle Scholar
  7. Cont R, Tankov P (2004) Financial modeling with jump processes. Chapman & Hall CRC Press, Boca RatonGoogle Scholar
  8. Devroye L (1986) Nonuniform random variate generation. Springer, New YorkGoogle Scholar
  9. Embrechts P, Maejima M (2002) Selfsimilar processes. Princeton University Press, PrincetonMATHGoogle Scholar
  10. Feller W (1971) Introduction to probability theory and its applications, vol II. Wiley, New YorkMATHGoogle Scholar
  11. Halgreen C (1979) Self-decomposability of the generalized inverse Gaussian and hyperbolic distributions. Z Wahrscheinlichkeitstheor Verw Geb 47:13–17MathSciNetMATHCrossRefGoogle Scholar
  12. Heyde CC (1999) A risky asset model with strong dependence through fractal activity time. J Appl Probab 34(4):1234–1239MathSciNetGoogle Scholar
  13. Heyde CC (2002) On modes of long-range dependence. J Appl Probab 39:882–888MathSciNetMATHCrossRefGoogle Scholar
  14. Heyde CC, Leonenko NN (2005) Student processes. Adv Appl Probab 37:342–365MathSciNetMATHCrossRefGoogle Scholar
  15. Jørgensen B (1982) Statistical properties of the generalized inverse Gaussian distribution. Lecture Notes in Statistics, vol 9. Springer-Verlag, New YorkGoogle Scholar
  16. Kelker D (1971) Infinite divisibility and variance mixtures of the normal distribution. Ann Math Statist 42:802–808MathSciNetMATHCrossRefGoogle Scholar
  17. Kozubowski TJ, Meerschaert MM, Podgorski K (2006) Fractional Laplace motion. Adv Appl Prob 38:451–464MathSciNetMATHCrossRefGoogle Scholar
  18. Linde W, Shi Z (2004) Evaluating the small deviation probabilities for subordinated Levy processes. Stoch Process Their Appl 113:273–287MathSciNetMATHCrossRefGoogle Scholar
  19. Madan DB, Seneta E (1990) The variance gamma (V.G.) model for share markets returns. J Bus 63:511–524CrossRefGoogle Scholar
  20. Madan DB, Carr P, Chang EC (1998) The variance gamma process and option pricing. European Finance Review 2:74–105CrossRefGoogle Scholar
  21. Mandelbrot BB (2001) Scaling in financial prices: I. Tails and dependence. Quantitative Finance 1:113–123MathSciNetCrossRefGoogle Scholar
  22. Mandelbrot BB, Taylor H (1967) On the distribution of stock price differences. Oper Res 15:1057–1062CrossRefGoogle Scholar
  23. Mandelbrot BB, Van Ness JW (1968) Fractional Brownian motion, fractional noises and applications. SIAM Rev 10:422–438MathSciNetMATHCrossRefGoogle Scholar
  24. Mandelbrot BB, Fisher A, Calvet L (1997) A multifractal model of asset returns. Cowles Foundation discussion paper no. 1164Google Scholar
  25. Meerschaert MM, Kozubowski TJ, Molz FJ, Lu S (2004) Fractional Laplace model for hydraulic conductivity. Geophys Res Lett 31:L08501CrossRefGoogle Scholar
  26. Molz FJ, Bowman GK (1993) A fractal-based stochastic interpolation scheme in subsurface hydrology. Water Resour Res 32:1183–1195Google Scholar
  27. Painter S (1996) Evidence for non-Gaussian scaling behavior in heterogeneous sedimentary formations. Water Resour Res 32:1183–1195CrossRefGoogle Scholar
  28. Perrin E, Harba R, Berzin-Joseph C, Iribarren I, Bonami A (2001) nth-Order fractional Brownian motion and fractional Gaussian noises. IEEE Trans Signal Process 49:1049–1059CrossRefGoogle Scholar
  29. Samorodnitsky G, Taqqu MS (2000) Stable non-Gaussian random processes: stochastic models with infinite variance. CRC Press, Boca RatonGoogle Scholar
  30. Sato K (2001) Subordination and self-decomposability. Stat Probab Lett 54(3):317–324MATHCrossRefGoogle Scholar
  31. Shephard N (1995) Statistical aspects of ARCH and stochastic volatility. In time series models. In: Cox DR, Hinkley DV, Barndorff-Nielsen OE (eds) Econometrics, finance and others fields. Chapman & Hall, London, pp 1–67Google Scholar
  32. Steutel FW, Van Harn K (2004) Infinite divisibility of probability distributions on the real line. Marcel Dekker, New YorkMATHGoogle Scholar
  33. Thorin O (1978) An extension of the notion of a generalized gamma convolution. Scand Actuar J 3:141–149MathSciNetCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology BombayMumbaiIndia

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