Modeling and analysis of financial time series beyond geometric Brownian motion

In this short review, the modeling and analysis of stock price financial series are presented in two different flavors: the dynamical picture provided by stochastic volatility models, capturing the non constant nature of price fluctuations, and a clustering Bayesian approach eliciting the underlying partition structure of data. The main theoretical results, obtained for specific mod- els, are presented and some examples of empirical analysis and financial application are then considered, showing the effectiveness of these approaches in capturing the non Gaussian behav- ior of empirical returns, their non trivial correlations, and, at a higher level, the effects of these features in determining the market risk exposure or the prices of stock option contracts.