Laws and Likelihoods for Ornstein Uhlenbeck-Gamma and other BNS OU Stochastic Volatilty models with extensions

In recent years there have been many proposals as flexible alternatives to Gaussian based continuous time stochastic volatility models. A great deal of these models employ positive L\'evy processes. Among these are the attractive non-Gaussian positive Ornstein-Uhlenbeck (OU) processes proposed by Barndorff-Nielsen and Shephard (BNS) in a series of papers. One current problem of these approaches is the unavailability of a tractable likelihood based statistical analysis for the returns of financial assets. This paper, while focusing on the BNS models, develops general theory for the implementation of statistical inference for a host of models. Specifically we show how to reduce the infinite-dimensional process based models to finite, albeit high, dimensional ones. Inference can then be based on Monte Carlo methods. As highlights, specific to BNS we show that an OU process driven by an infinite activity Gamma process, that is an OU-$\Gamma$, exhibits unique features which allows one to exactly sample from relevant joint distributions. We show that this is a consequence of the OU structure and the unique calculus of Gamma and Dirichlet processes. Owing to another connection between Gamma/Dirichlet processes and the theory of Generalized Gamma Convolutions (GGC) we identify a large class of models, we call (FGGC), where one can perfectly sample marginal distributions relevant to option pricing and Monte Carlo likelihood analysis. This involves a curious result, we establish as Theorem 6.1. We also discuss analytic techniques and candidate densities for Monte-Carlo procedures which apply more generally.