  In this paper we describe three stochastic models based on a semi-Markov
chains approach and its generalizations to study the high frequency price
dynamics of traded stocks. The three models are: a simple semi-Markov chain
model, an indexed semi-Markov chain model and a weighted indexed semi-Markov
chain model. We show, through Monte Carlo simulations, that the models are able
to reproduce important stylized facts of financial time series as the
persistence of volatility. In particular, we analyzed high frequency data from
the Italian stock market from the first of January 2007 until end of December
2010 and we apply to it the semi-Markov chain model and the indexed semi-Markov
chain model. The last model, instead, is applied to data from Italian and
German stock markets from January 1, 2007 until the end of December 2010.
