Fractional Normal Inverse Gaussian Process

TL;DR

This paper introduces the fractional normal inverse Gaussian (FNIG) process, a new stochastic process with dependent increments, heavy tails, and long-range dependence, extending the classical NIG process by incorporating fractional Brownian motion.

Contribution

It proposes the FNIG process by subordinating fractional Brownian motion to an inverse Gaussian process, and discusses its properties and potential applications.

Findings

FNIG process has heavy-tailed marginal distributions.

First order increments of FNIG exhibit long-range dependence.

The process shows persistence of signs LRD property.

Abstract

Normal inverse Gaussian (NIG) process was introduced by Barndorff-Nielsen (1997) by subordinating Brownian motion with drift to an inverse Gaussian process. Increments of NIG process are independent and 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 of the FNIG process called 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