Fractional Normal Inverse Gaussian Process

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.