  We review statistical properties of models generated by the application of a
(positive and negative order) fractional derivative operator to a standard
random walk and show that the resulting stochastic walks display
slowly-decaying autocorrelation functions. The relation between these
correlated walks and the well-known fractionally integrated autoregressive
(FIGARCH) models, commonly used in econometric studies, is discussed. The
application of correlated random walks to simulate empirical financial times
series is considered and compared with the predictions from FIGARCH and the
simpler FIARCH processes. A comparison with empirical data is performed.
