Generalized hyperbolic and inverse Gaussian distributions: Limiting cases and approximation of processes

In recent years more realistic stochastic models for price movements in financial markets have been developed by replacing the classical Brownian motion by Levy processes. Among these generalized hyperbolic Levy processes turned out to provide an excellent fit to observed market data. We review the most important facts of the generating distributions and give a detailed derivation of its limits including some well known distributions which also have been applied to financial data. All these are subclasses of extended generalized Gamma-convolutions. This fact allows the explicit calculation of characteristic triplets and the construction of uan convergent triangular schemes, whereas classical multinomial approximations are shown to fail in this context. The mixing generalized inverse Gaussian distributions are also considered.