Beyond Brownian Motion
Daniel J. Duffy, Jörg Kienitz · 2012
This chapter discusses Lévy processes and examines the class of Normal Mean Variance Mixture models, which includes the Normal Inverse Gaussian (NIG) and Variance Gamma (VG) processes. Normal mean variance mixture models generalize the multivariate normal distribution in the sense that the mean and the variance are determined by random variables. The idea is to introduce another source of randomness into the covariance matrix and then the mean via a mixing variable. The chapter discusses the main properties of Normal Mean Variance Mixture models and shows that simulating these processes is possible by subordinating a standard Brownian motion. Implementation in C++ is presented and some examples on how to implement such models are discussed, for which the NIG and VG processes as well as a multi-dimensional version of the NIG model are considered.