NORMAL INVERSE GAUSSIAN DISTRIBUTIONS AND STOCHASTIC VOLATILITY MODELING

The normal inverse Gaussian distribution is defined as a variance-mean mixture of a normal distribution with the inverse Gaussian as the mix ing distribution. The distribution determines an homogeneous Levy proc ess, and this process is representable through subordination of Browni an motion by the inverse Gaussian process. The canonical, Levy type, d ecomposition of the process is determined, As a preparation for develo pments in the latter part of the paper the connection of the normal in verse Gaussian distribution to the classes of generalized hyperbolic a nd inverse Gaussian distributions is briefly reviewed. Then a discussi on is begun of the potential of the normal inverse Gaussian distributi on and Levy process for modelling and analysing statistical data, with particular reference to extensive sets of observations from turbulenc e and from finance. These areas of application imply a need for extend ing the inverse Gaussian Levy process so as to accommodate certain, fr equently observed, temporal dependence structures. Some extensions, of the stochastic volatility type, are constructed via an observation-dr iven approach to state space modelling, At the end of the paper genera lizations to multivariate settings are indicated.