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Fractional Normal Inverse Gaussian Process

  • Arun Kumar
  • Palaniappan Vellaisamy
Article

Abstract

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.

Keywords

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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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology BombayMumbaiIndia

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