Abstract: The currently most popular models for the volatility of financial time series, non Gaussian Ornstein-Uhlenbeck stochastic processes are extended to more general non Ornstein-Uhlenbeck models driven by the general Lévy process, such as Generalized Inverse Gaussian(GIG)and Tempered Stable distributions(CGMY). In particular, means of making the correlation structure in the volatility process more flexible based on continuous superpositions of the more general non Ornstein-Uhlenbeck models are investigated, which can introduce long-memory into the volatility model. A shot-noise process and approximation for the continuous superpositions process are represented. Inference is carried out in a Bayesian framework, with computation using extended Reversible Jump Markov chain Monte Carlo and dependent thinning to the continuous superposition case. Empirical research demonstrates that the efficient Markov chain Monte Carlo methods appear to be successful in the case of the general GIG and CGMY marginal model, and that those models can be fitted to real share price returns data, and that results indicate that for the series we study, the long-memory aspect of the model is supported.

Key words: Lévy process, no Gaussian Ornstein-Uhlenbeck stochastic processes, reversible jump Markov chain Monte Carlo, long-memory